Poisson's ratio - Wikipedia

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cdx-button--icon-only vector-toc-toggle"> Toggle Poisson's ratio from geometry changes subsection </button> <ul id="toc-Poisson's_ratio_from_geometry_changes-sublist" class="vector-toc-list"> <li id="toc-Length_change" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Length_change"> <div class="vector-toc-text"> 2.1 Length change </div> </a> <ul id="toc-Length_change-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Volumetric_change" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Volumetric_change"> <div class="vector-toc-text"> 2.2 Volumetric change </div> </a> <ul id="toc-Volumetric_change-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Width_change" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Width_change"> <div class="vector-toc-text"> 2.3 Width change </div> </a> <ul id="toc-Width_change-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Characteristic_materials" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Characteristic_materials"> <div class="vector-toc-text"> 3 Characteristic materials </div> </a> <button aria-controls="toc-Characteristic_materials-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Characteristic materials subsection </button> <ul id="toc-Characteristic_materials-sublist" class="vector-toc-list"> <li id="toc-Isotropic" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Isotropic"> <div class="vector-toc-text"> 3.1 Isotropic </div> </a> <ul id="toc-Isotropic-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Anisotropic" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Anisotropic"> <div class="vector-toc-text"> 3.2 Anisotropic </div> </a> <ul id="toc-Anisotropic-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Orthotropic" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Orthotropic"> <div class="vector-toc-text"> 3.3 Orthotropic </div> </a> <ul id="toc-Orthotropic-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Transversely_isotropic" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Transversely_isotropic"> <div class="vector-toc-text"> 3.4 Transversely isotropic </div> </a> <ul id="toc-Transversely_isotropic-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Poisson's_ratio_values_for_different_materials" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Poisson's_ratio_values_for_different_materials"> <div class="vector-toc-text"> 4 Poisson's ratio values for different materials </div> </a> <button aria-controls="toc-Poisson's_ratio_values_for_different_materials-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Poisson's ratio values for different materials subsection </button> <ul id="toc-Poisson's_ratio_values_for_different_materials-sublist" class="vector-toc-list"> <li id="toc-Negative_Poisson's_ratio_materials" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Negative_Poisson's_ratio_materials"> <div class="vector-toc-text"> 4.1 Negative Poisson's ratio materials </div> </a> <ul id="toc-Negative_Poisson's_ratio_materials-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Poisson_function" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Poisson_function"> <div class="vector-toc-text"> 5 Poisson function </div> </a> <ul id="toc-Poisson_function-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Applications_of_Poisson's_effect" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Applications_of_Poisson's_effect"> <div class="vector-toc-text"> 6 Applications of Poisson's effect </div> </a> <ul id="toc-Applications_of_Poisson's_effect-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> 7 See also </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> 8 References </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> 9 External links </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > Toggle the table of contents </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading">Poisson's ratio</h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 56 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-56" aria-hidden="true" > 56 languages </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Poisson_se_verhouding" title="Poisson se verhouding – Afrikaans" lang="af" hreflang="af" data-title="Poisson se verhouding" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target">Afrikaans</a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%86%D8%B3%D8%A8%D8%A9_%D8%A8%D9%88%D8%A7%D8%B3%D9%88%D9%86" title="نسبة بواسون – Arabic" lang="ar" hreflang="ar" data-title="نسبة بواسون" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-as mw-list-item"><a href="https://as.wikipedia.org/wiki/%E0%A6%AA%E0%A6%AF%E0%A6%BC%E0%A6%9B%E0%A6%A8%E0%A7%B0_%E0%A6%85%E0%A6%A8%E0%A7%81%E0%A6%AA%E0%A6%BE%E0%A6%A4" title="পয়ছনৰ অনুপাত – Assamese" lang="as" hreflang="as" data-title="পয়ছনৰ অনুপাত" data-language-autonym="অসমীয়া" data-language-local-name="Assamese" class="interlanguage-link-target">অসমীয়া</a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Coeficiente_de_Poisson" title="Coeficiente de Poisson – Asturian" lang="ast" hreflang="ast" data-title="Coeficiente de Poisson" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target">Asturianu</a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AA%E0%A6%AF%E0%A6%BC%E0%A6%B8%E0%A6%A8%E0%A7%87%E0%A6%B0_%E0%A6%85%E0%A6%A8%E0%A7%81%E0%A6%AA%E0%A6%BE%E0%A6%A4" title="পয়সনের অনুপাত – Bangla" lang="bn" hreflang="bn" data-title="পয়সনের অনুপাত" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target">বাংলা</a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9A%D0%B0%D1%8D%D1%84%D1%96%D1%86%D1%8B%D0%B5%D0%BD%D1%82_%D0%9F%D1%83%D0%B0%D1%81%D0%BE%D0%BD%D0%B0" title="Каэфіцыент Пуасона – Belarusian" lang="be" hreflang="be" data-title="Каэфіцыент Пуасона" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target">Беларуская</a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%9A%D0%B0%D1%8D%D1%84%D1%96%D1%86%D1%8B%D0%B5%D0%BD%D1%82_%D0%9F%D1%83%D0%B0%D1%81%D0%BE%D0%BD%D0%B0" title="Каэфіцыент Пуасона – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Каэфіцыент Пуасона" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target">Беларуская (тарашкевіца)</a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9A%D0%BE%D0%B5%D1%84%D0%B8%D1%86%D0%B8%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%9F%D0%BE%D0%B0%D1%81%D0%BE%D0%BD" title="Коефициент на Поасон – Bulgarian" lang="bg" hreflang="bg" data-title="Коефициент на Поасон" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target">Български</a></li><li class="interlanguage-link interwiki-bar mw-list-item"><a href="https://bar.wikipedia.org/wiki/Queakauntroktiaunszoih" title="Queakauntroktiaunszoih – Bavarian" lang="bar" hreflang="bar" data-title="Queakauntroktiaunszoih" data-language-autonym="Boarisch" data-language-local-name="Bavarian" class="interlanguage-link-target">Boarisch</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Coeficient_de_Poisson" title="Coeficient de Poisson – Catalan" lang="ca" hreflang="ca" data-title="Coeficient de Poisson" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Poissonova_konstanta_(mechanika)" title="Poissonova konstanta (mechanika) – Czech" lang="cs" hreflang="cs" data-title="Poissonova konstanta (mechanika)" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Poissons_forhold" title="Poissons forhold – Danish" lang="da" hreflang="da" data-title="Poissons forhold" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target">Dansk</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Poissonzahl" title="Poissonzahl – German" lang="de" hreflang="de" data-title="Poissonzahl" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Poissoni_tegur" title="Poissoni tegur – Estonian" lang="et" hreflang="et" data-title="Poissoni tegur" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target">Eesti</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Coeficiente_de_Poisson" title="Coeficiente de Poisson – Spanish" lang="es" hreflang="es" data-title="Coeficiente de Poisson" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Rilatumo_de_Poisson" title="Rilatumo de Poisson – Esperanto" lang="eo" hreflang="eo" data-title="Rilatumo de Poisson" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target">Esperanto</a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Poissonen_koefiziente" title="Poissonen koefiziente – Basque" lang="eu" hreflang="eu" data-title="Poissonen koefiziente" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target">Euskara</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%86%D8%B3%D8%A8%D8%AA_%D9%BE%D9%88%D8%A7%D8%B3%D9%88%D9%86" title="نسبت پواسون – Persian" lang="fa" hreflang="fa" data-title="نسبت پواسون" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Coefficient_de_Poisson" title="Coefficient de Poisson – French" lang="fr" hreflang="fr" data-title="Coefficient de Poisson" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/C%C3%B3imheas_Poisson" title="Cóimheas Poisson – Irish" lang="ga" hreflang="ga" data-title="Cóimheas Poisson" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target">Gaeilge</a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Coeficiente_de_Poisson" title="Coeficiente de Poisson – Galician" lang="gl" hreflang="gl" data-title="Coeficiente de Poisson" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target">Galego</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%91%B8%EC%95%84%EC%86%A1_%EB%B9%84" title="푸아송 비 – Korean" lang="ko" hreflang="ko" data-title="푸아송 비" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%8A%D5%B8%D6%82%D5%A1%D5%BD%D5%B8%D5%B6%D5%AB_%D5%A3%D5%B8%D6%80%D5%AE%D5%A1%D5%AF%D5%AB%D6%81" title="Պուասոնի գործակից – Armenian" lang="hy" hreflang="hy" data-title="Պուասոնի գործակից" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target">Հայերեն</a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AA%E0%A5%8D%E0%A4%B5%E0%A4%BE%E0%A4%B8%E0%A5%8B%E0%A4%82_%E0%A4%85%E0%A4%A8%E0%A5%81%E0%A4%AA%E0%A4%BE%E0%A4%A4" title="प्वासों अनुपात – Hindi" lang="hi" hreflang="hi" data-title="प्वासों अनुपात" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target">हिन्दी</a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Poissonov_omjer" title="Poissonov omjer – Croatian" lang="hr" hreflang="hr" data-title="Poissonov omjer" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target">Hrvatski</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Rasio_Poisson" title="Rasio Poisson – Indonesian" lang="id" hreflang="id" data-title="Rasio Poisson" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Coefficiente_di_Poisson" title="Coefficiente di Poisson – Italian" lang="it" hreflang="it" data-title="Coefficiente di Poisson" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A7%D7%93%D7%9D_%D7%A4%D7%95%D7%90%D7%A1%D7%95%D7%9F" title="מקדם פואסון – Hebrew" lang="he" hreflang="he" data-title="מקדם פואסון" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%9E%E1%83%A3%E1%83%90%E1%83%A1%E1%83%9D%E1%83%9C%E1%83%98%E1%83%A1_%E1%83%99%E1%83%9D%E1%83%94%E1%83%A4%E1%83%98%E1%83%AA%E1%83%98%E1%83%94%E1%83%9C%E1%83%A2%E1%83%98" title="პუასონის კოეფიციენტი – Georgian" lang="ka" hreflang="ka" data-title="პუასონის კოეფიციენტი" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target">ქართული</a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9F%D1%83%D0%B0%D1%81%D1%81%D0%BE%D0%BD_%D0%BA%D0%BE%D1%8D%D1%84%D1%84%D0%B8%D1%86%D0%B8%D0%B5%D0%BD%D1%82%D1%96" title="Пуассон коэффициенті – Kazakh" lang="kk" hreflang="kk" data-title="Пуассон коэффициенті" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target">Қазақша</a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Puasono_santykis" title="Puasono santykis – Lithuanian" lang="lt" hreflang="lt" data-title="Puasono santykis" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target">Lietuvių</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Poisson-t%C3%A9nyez%C5%91" title="Poisson-tényező – Hungarian" lang="hu" hreflang="hu" data-title="Poisson-tényező" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%AA%E0%B5%8B%E0%B4%AF%E0%B5%8D%E0%B4%B8%E0%B5%BA_%E0%B4%85%E0%B4%A8%E0%B5%81%E0%B4%AA%E0%B4%BE%E0%B4%A4%E0%B4%82" title="പോയ്സൺ അനുപാതം – Malayalam" lang="ml" hreflang="ml" data-title="പോയ്സൺ അനുപാതം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target">മലയാളം</a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Nisbah_Poisson" title="Nisbah Poisson – Malay" lang="ms" hreflang="ms" data-title="Nisbah Poisson" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target">Bahasa Melayu</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Poisson-factor" title="Poisson-factor – Dutch" lang="nl" hreflang="nl" data-title="Poisson-factor" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%9D%E3%82%A2%E3%82%BD%E3%83%B3%E6%AF%94" title="ポアソン比 – Japanese" lang="ja" hreflang="ja" data-title="ポアソン比" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Poissontalet" title="Poissontalet – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Poissontalet" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target">Norsk nynorsk</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Liczba_Poissona" title="Liczba Poissona – Polish" lang="pl" hreflang="pl" data-title="Liczba Poissona" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Coeficiente_de_Poisson" title="Coeficiente de Poisson – Portuguese" lang="pt" hreflang="pt" data-title="Coeficiente de Poisson" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Coeficientul_lui_Poisson" title="Coeficientul lui Poisson – Romanian" lang="ro" hreflang="ro" data-title="Coeficientul lui Poisson" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target">Română</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9A%D0%BE%D1%8D%D1%84%D1%84%D0%B8%D1%86%D0%B8%D0%B5%D0%BD%D1%82_%D0%9F%D1%83%D0%B0%D1%81%D1%81%D0%BE%D0%BD%D0%B0" title="Коэффициент Пуассона – Russian" lang="ru" hreflang="ru" data-title="Коэффициент Пуассона" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Poisson%27s_ratio" title="Poisson's ratio – Simple English" lang="en-simple" hreflang="en-simple" data-title="Poisson's ratio" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target">Simple English</a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Poissonova_kon%C5%A1tanta_(mechanika)" title="Poissonova konštanta (mechanika) – Slovak" lang="sk" hreflang="sk" data-title="Poissonova konštanta (mechanika)" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target">Slovenčina</a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Poissonovo_%C5%A1tevilo" title="Poissonovo število – Slovenian" lang="sl" hreflang="sl" data-title="Poissonovo število" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target">Slovenščina</a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/Poasonov_odnos" title="Poasonov odnos – Serbian" lang="sr" hreflang="sr" data-title="Poasonov odnos" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target">Српски / srpski</a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Poissonov_omjer" title="Poissonov omjer – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Poissonov omjer" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target">Srpskohrvatski / српскохрватски</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Poissonin_suhde" title="Poissonin suhde – Finnish" lang="fi" hreflang="fi" data-title="Poissonin suhde" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Poissons_konstant" title="Poissons konstant – Swedish" lang="sv" hreflang="sv" data-title="Poissons konstant" data-language-autonym="Svenska" data-language-local-name="Swedish" 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href="https://tr.wikipedia.org/wiki/Poisson_oran%C4%B1" title="Poisson oranı – Turkish" lang="tr" hreflang="tr" data-title="Poisson oranı" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target">Türkçe</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9A%D0%BE%D0%B5%D1%84%D1%96%D1%86%D1%96%D1%94%D0%BD%D1%82_%D0%9F%D1%83%D0%B0%D1%81%D1%81%D0%BE%D0%BD%D0%B0" title="Коефіцієнт Пуассона – Ukrainian" lang="uk" hreflang="uk" data-title="Коефіцієнт Пуассона" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target">Українська</a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/H%E1%BB%87_s%E1%BB%91_Poisson" title="Hệ số Poisson – Vietnamese" lang="vi" hreflang="vi" data-title="Hệ số Poisson" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" 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typeof="mw:File/Thumb"><a href="/wiki/File:Poisson_ratio_compression_example.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/67/Poisson_ratio_compression_example.svg/220px-Poisson_ratio_compression_example.svg.png" decoding="async" width="220" height="296" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/67/Poisson_ratio_compression_example.svg/330px-Poisson_ratio_compression_example.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/67/Poisson_ratio_compression_example.svg/440px-Poisson_ratio_compression_example.svg.png 2x" data-file-width="215" data-file-height="289" /></a><figcaption>Poisson's ratio of a material defines the ratio of transverse strain (x direction) to the axial strain (y direction)</figcaption></figure> In <a href="/wiki/Materials_science" title="Materials science">materials science</a> and <a href="/wiki/Solid_mechanics" title="Solid mechanics">solid mechanics</a>, Poisson's ratio (symbol: ν (<a href="/wiki/Nu_(letter)" title="Nu (letter)">nu</a>)) is a measure of the Poisson effect, the <a href="/wiki/Deformation_(engineering)" title="Deformation (engineering)">deformation</a> (expansion or contraction) of a material in directions perpendicular to the specific direction of <a href="/wiki/Structural_load" title="Structural load">loading</a>. The value of Poisson's ratio is the negative of the ratio of <a href="/wiki/Lateral_strain" title="Lateral strain">transverse strain</a> to axial <a href="/wiki/Strain_(materials_science)" class="mw-redirect" title="Strain (materials science)">strain</a>. For small values of these changes, ν is the amount of transversal <a href="/wiki/Elongation_(materials_science)" class="mw-redirect" title="Elongation (materials science)">elongation</a> divided by the amount of axial <a href="/wiki/Compressive_strength" title="Compressive strength">compression</a>. Most materials have Poisson's ratio values ranging between 0.0 and 0.5. For soft materials,<a href="#cite_note-1">[1]</a> such as rubber, where the bulk modulus is much higher than the shear modulus, Poisson's ratio is near 0.5. For open-cell polymer foams, Poisson's ratio is near zero, since the cells tend to collapse in compression. Many typical solids have Poisson's ratios in the range of 0.2 to 0.3. The ratio is named after the French mathematician and physicist <a href="/wiki/Sim%C3%A9on_Poisson" class="mw-redirect" title="Siméon Poisson">Siméon Poisson</a>. <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Origin">Origin</h2>[<a href="/w/index.php?title=Poisson%27s_ratio&action=edit&section=1" title="Edit section: Origin">edit</a>]</div> Poisson's ratio is a measure of the Poisson effect, the phenomenon in which a material tends to expand in directions perpendicular to the direction of compression. Conversely, if the material is stretched rather than compressed, it usually tends to contract in the directions transverse to the direction of stretching. It is a common observation when a rubber band is stretched, it becomes noticeably thinner. Again, the Poisson ratio will be the ratio of relative contraction to relative expansion and will have the same value as above. In certain rare cases,<a href="#cite_note-lakes-2">[2]</a> a material will actually shrink in the transverse direction when compressed (or expand when stretched) which will yield a negative value of the Poisson ratio. The Poisson's ratio of a stable, <a href="/wiki/Isotropy#Materials_science" title="Isotropy">isotropic</a>, linear <a href="/wiki/Elasticity_(physics)" title="Elasticity (physics)">elastic</a> material must be between −1.0 and +0.5 because of the requirement for <a href="/wiki/Young%27s_modulus" title="Young's modulus">Young's modulus</a>, the <a href="/wiki/Shear_modulus" title="Shear modulus">shear modulus</a> and <a href="/wiki/Bulk_modulus" title="Bulk modulus">bulk modulus</a> to have positive values.<a href="#cite_note-3">[3]</a> Most materials have Poisson's ratio values ranging between 0.0 and 0.5. A perfectly incompressible isotropic material deformed elastically at small strains would have a Poisson's ratio of exactly 0.5. Most steels and rigid polymers when used within their design limits (before <a href="/wiki/Yield_(engineering)" title="Yield (engineering)">yield</a>) exhibit values of about 0.3, increasing to 0.5 for post-yield deformation which occurs largely at constant volume.<a href="#cite_note-4">[4]</a> Rubber has a Poisson ratio of nearly 0.5. Cork's Poisson ratio is close to 0, showing very little lateral expansion when compressed and glass is between 0.18 and 0.30. Some materials, e.g. some polymer foams, origami folds,<a href="#cite_note-5">[5]</a><a href="#cite_note-6">[6]</a> and certain cells can exhibit negative Poisson's ratio, and are referred to as <a href="/wiki/Auxetics" title="Auxetics">auxetic materials</a>. If these auxetic materials are stretched in one direction, they become thicker in the perpendicular direction. In contrast, some <a href="/wiki/Anisotropy" title="Anisotropy">anisotropic</a> materials, such as <a href="/wiki/Carbon_nanotube" title="Carbon nanotube">carbon nanotubes</a>, zigzag-based folded sheet materials,<a href="#cite_note-:0-7">[7]</a><a href="#cite_note-8">[8]</a> and honeycomb auxetic metamaterials<a href="#cite_note-9">[9]</a> to name a few, can exhibit one or more Poisson's ratios above 0.5 in certain directions. Assuming that the material is stretched or compressed in only one direction (the x axis in the diagram below): <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu =-{\frac {d\varepsilon _{\mathrm {trans} }}{d\varepsilon _{\mathrm {axial} }}}=-{\frac {d\varepsilon _{\mathrm {y} }}{d\varepsilon _{\mathrm {x} }}}=-{\frac {d\varepsilon _{\mathrm {z} }}{d\varepsilon _{\mathrm {x} }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">x</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">l</mi> </mrow> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">y</mi> </mrow> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">x</mi> </mrow> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">z</mi> </mrow> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">x</mi> </mrow> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu =-{\frac {d\varepsilon _{\mathrm {trans} }}{d\varepsilon _{\mathrm {axial} }}}=-{\frac {d\varepsilon _{\mathrm {y} }}{d\varepsilon _{\mathrm {x} }}}=-{\frac {d\varepsilon _{\mathrm {z} }}{d\varepsilon _{\mathrm {x} }}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4854a5e8d3492c57a7a8141cf5edb2e8ea148048" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:31.459ex; height:6.176ex;" alt="{\displaystyle \nu =-{\frac {d\varepsilon _{\mathrm {trans} }}{d\varepsilon _{\mathrm {axial} }}}=-{\frac {d\varepsilon _{\mathrm {y} }}{d\varepsilon _{\mathrm {x} }}}=-{\frac {d\varepsilon _{\mathrm {z} }}{d\varepsilon _{\mathrm {x} }}}}"></dd></dl> where <ul><li>ν is the resulting Poisson's ratio,</li> <li>εtrans is transverse strain</li> <li>εaxial is axial strain</li></ul> and positive strain indicates extension and negative strain indicates contraction. <div class="mw-heading mw-heading2"><h2 id="Poisson's_ratio_from_geometry_changes">Poisson's ratio from geometry changes</h2>[<a href="/w/index.php?title=Poisson%27s_ratio&action=edit&section=2" title="Edit section: Poisson's ratio from geometry changes">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Length_change">Length change</h3>[<a href="/w/index.php?title=Poisson%27s_ratio&action=edit&section=3" title="Edit section: Length change">edit</a>]</div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:PoissonRatio.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/PoissonRatio.svg/300px-PoissonRatio.svg.png" decoding="async" width="300" height="297" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/PoissonRatio.svg/450px-PoissonRatio.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/PoissonRatio.svg/600px-PoissonRatio.svg.png 2x" data-file-width="588" data-file-height="583" /></a><figcaption>Figure 1: A cube with sides of length L of an isotropic linearly elastic material subject to tension along the x axis, with a Poisson's ratio of 0.5. The green cube is unstrained, the red is expanded in the x-direction by ΔL due to tension, and contracted in the y- and z-directions by ΔL′.</figcaption></figure> For a cube stretched in the x-direction (see Figure 1) with a length increase of ΔL in the x-direction, and a length decrease of ΔL′ in the y- and z-directions, the infinitesimal diagonal strains are given by <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\varepsilon _{x}={\frac {dx}{x}},\qquad d\varepsilon _{y}={\frac {dy}{y}},\qquad d\varepsilon _{z}={\frac {dz}{z}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>x</mi> </mrow> <mi>x</mi> </mfrac> </mrow> <mo>,</mo> <mspace width="2em" /> <mi>d</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>y</mi> </mrow> <mi>y</mi> </mfrac> </mrow> <mo>,</mo> <mspace width="2em" /> <mi>d</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>z</mi> </mrow> <mi>z</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\varepsilon _{x}={\frac {dx}{x}},\qquad d\varepsilon _{y}={\frac {dy}{y}},\qquad d\varepsilon _{z}={\frac {dz}{z}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28ff1adcc9eb4058b78e26c6c4bcf705a10b5c48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:41.151ex; height:5.843ex;" alt="{\displaystyle d\varepsilon _{x}={\frac {dx}{x}},\qquad d\varepsilon _{y}={\frac {dy}{y}},\qquad d\varepsilon _{z}={\frac {dz}{z}}.}"></dd></dl> If Poisson's ratio is constant through deformation, integrating these expressions and using the definition of Poisson's ratio gives <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\nu \int _{L}^{L+\Delta L}{\frac {dx}{x}}=\int _{L}^{L+\Delta L'}{\frac {dy}{y}}=\int _{L}^{L+\Delta L'}{\frac {dz}{z}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>ν</mi> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <mi>L</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>x</mi> </mrow> <mi>x</mi> </mfrac> </mrow> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <msup> <mi>L</mi> <mo>′</mo> </msup> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>y</mi> </mrow> <mi>y</mi> </mfrac> </mrow> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <msup> <mi>L</mi> <mo>′</mo> </msup> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>z</mi> </mrow> <mi>z</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\nu \int _{L}^{L+\Delta L}{\frac {dx}{x}}=\int _{L}^{L+\Delta L'}{\frac {dy}{y}}=\int _{L}^{L+\Delta L'}{\frac {dz}{z}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca412f87e75cf5da77b38125aba36b75cf07644f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:44.972ex; height:6.509ex;" alt="{\displaystyle -\nu \int _{L}^{L+\Delta L}{\frac {dx}{x}}=\int _{L}^{L+\Delta L'}{\frac {dy}{y}}=\int _{L}^{L+\Delta L'}{\frac {dz}{z}}.}"></dd></dl> Solving and exponentiating, the relationship between ΔL and ΔL′ is then <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(1+{\frac {\Delta L}{L}}\right)^{-\nu }=1+{\frac {\Delta L'}{L}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <mi>L</mi> </mrow> <mi>L</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>ν</mi> </mrow> </msup> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <msup> <mi>L</mi> <mo>′</mo> </msup> </mrow> <mi>L</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(1+{\frac {\Delta L}{L}}\right)^{-\nu }=1+{\frac {\Delta L'}{L}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b843970e4e08e3333d560bc58ad88b586a6ae75b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.949ex; height:6.509ex;" alt="{\displaystyle \left(1+{\frac {\Delta L}{L}}\right)^{-\nu }=1+{\frac {\Delta L'}{L}}.}"></dd></dl> For very small values of ΔL and ΔL′, the first-order approximation yields: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu \approx -{\frac {\Delta L'}{\Delta L}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> <mo>≈</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <msup> <mi>L</mi> <mo>′</mo> </msup> </mrow> <mrow> <mi mathvariant="normal">Δ</mi> <mi>L</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu \approx -{\frac {\Delta L'}{\Delta L}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6838e9cd7bfb89554b3d1b423f18cca40bd478fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.825ex; height:5.676ex;" alt="{\displaystyle \nu \approx -{\frac {\Delta L'}{\Delta L}}.}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Volumetric_change">Volumetric change</h3>[<a href="/w/index.php?title=Poisson%27s_ratio&action=edit&section=4" title="Edit section: Volumetric change">edit</a>]</div> The relative change of volume <style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style>⁠ΔV/V⁠ of a cube due to the stretch of the material can now be calculated. Since V = L3 and <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V+\Delta V=(L+\Delta L)\left(L+\Delta L'\right)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <mi>V</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>L</mi> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <mi>L</mi> <mo stretchy="false">)</mo> <msup> <mrow> <mo>(</mo> <mrow> <mi>L</mi> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <msup> <mi>L</mi> <mo>′</mo> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V+\Delta V=(L+\Delta L)\left(L+\Delta L'\right)^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d96599d365d3702cb0c380a3b3903ab7945713e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.691ex; height:3.509ex;" alt="{\displaystyle V+\Delta V=(L+\Delta L)\left(L+\Delta L'\right)^{2}}"></dd></dl> one can derive <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\Delta V}{V}}=\left(1+{\frac {\Delta L}{L}}\right)\left(1+{\frac {\Delta L'}{L}}\right)^{2}-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <mi>V</mi> </mrow> <mi>V</mi> </mfrac> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <mi>L</mi> </mrow> <mi>L</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <msup> <mi>L</mi> <mo>′</mo> </msup> </mrow> <mi>L</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\Delta V}{V}}=\left(1+{\frac {\Delta L}{L}}\right)\left(1+{\frac {\Delta L'}{L}}\right)^{2}-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1abb25d2acfb7e0358b642a4b0c1052df06d9a84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:37.344ex; height:6.509ex;" alt="{\displaystyle {\frac {\Delta V}{V}}=\left(1+{\frac {\Delta L}{L}}\right)\left(1+{\frac {\Delta L'}{L}}\right)^{2}-1}"></dd></dl> Using the above derived relationship between ΔL and ΔL′: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\Delta V}{V}}=\left(1+{\frac {\Delta L}{L}}\right)^{1-2\nu }-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <mi>V</mi> </mrow> <mi>V</mi> </mfrac> </mrow> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <mi>L</mi> </mrow> <mi>L</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mi>ν</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\Delta V}{V}}=\left(1+{\frac {\Delta L}{L}}\right)^{1-2\nu }-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a86ed2906ffcda23a462d51f2cb306741826c4c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:27.465ex; height:6.509ex;" alt="{\displaystyle {\frac {\Delta V}{V}}=\left(1+{\frac {\Delta L}{L}}\right)^{1-2\nu }-1}"></dd></dl> and for very small values of ΔL and ΔL′, the first-order approximation yields: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\Delta V}{V}}\approx (1-2\nu ){\frac {\Delta L}{L}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <mi>V</mi> </mrow> <mi>V</mi> </mfrac> </mrow> <mo>≈</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mi>ν</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <mi>L</mi> </mrow> <mi>L</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\Delta V}{V}}\approx (1-2\nu ){\frac {\Delta L}{L}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc607c039c775dec829347bcd7e928f2fa4118ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:20.219ex; height:5.509ex;" alt="{\displaystyle {\frac {\Delta V}{V}}\approx (1-2\nu ){\frac {\Delta L}{L}}}"></dd></dl> For isotropic materials we can use <a href="/wiki/Lam%C3%A9_parameters" title="Lamé parameters">Lamé's relation</a><a href="#cite_note-10">[10]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu \approx {\frac {1}{2}}-{\frac {E}{6K}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>E</mi> <mrow> <mn>6</mn> <mi>K</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu \approx {\frac {1}{2}}-{\frac {E}{6K}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d273038934e9c555e91bf84ff37998ab1c9a89c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:13.234ex; height:5.343ex;" alt="{\displaystyle \nu \approx {\frac {1}{2}}-{\frac {E}{6K}}}"></dd></dl> where K is <a href="/wiki/Bulk_modulus" title="Bulk modulus">bulk modulus</a> and E is <a href="/wiki/Young%27s_modulus" title="Young's modulus">Young's modulus</a>. <div class="mw-heading mw-heading3"><h3 id="Width_change">Width change</h3>[<a href="/w/index.php?title=Poisson%27s_ratio&action=edit&section=5" title="Edit section: Width change">edit</a>]</div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Rod_diameter_change_poisson.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Rod_diameter_change_poisson.svg/350px-Rod_diameter_change_poisson.svg.png" decoding="async" width="350" height="338" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Rod_diameter_change_poisson.svg/525px-Rod_diameter_change_poisson.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Rod_diameter_change_poisson.svg/700px-Rod_diameter_change_poisson.svg.png 2x" data-file-width="444" data-file-height="429" /></a><figcaption>Figure 2: The blue slope represents a simplified formula (the top one in the legend) that works well for modest deformations, ∆L, up to about ±3. The green curve represents a formula better suited for larger deformations.</figcaption></figure> If a rod with diameter (or width, or thickness) d and length L is subject to tension so that its length will change by ΔL then its diameter d will change by: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\Delta d}{d}}=-\nu {\frac {\Delta L}{L}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <mi>d</mi> </mrow> <mi>d</mi> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <mi>L</mi> </mrow> <mi>L</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\Delta d}{d}}=-\nu {\frac {\Delta L}{L}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c31146df35fd9579a5cdee091c6d9839205b3705" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.481ex; height:5.509ex;" alt="{\displaystyle {\frac {\Delta d}{d}}=-\nu {\frac {\Delta L}{L}}}"></dd></dl> The above formula is true only in the case of small deformations; if deformations are large then the following (more precise) formula can be used: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta d=-d\left(1-{\left(1+{\frac {\Delta L}{L}}\right)}^{-\nu }\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>d</mi> <mo>=</mo> <mo>−</mo> <mi>d</mi> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <mi>L</mi> </mrow> <mi>L</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>ν</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta d=-d\left(1-{\left(1+{\frac {\Delta L}{L}}\right)}^{-\nu }\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b36c33ea0ee14a87f8da9f572e8cb37a50e496e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:31.506ex; height:7.509ex;" alt="{\displaystyle \Delta d=-d\left(1-{\left(1+{\frac {\Delta L}{L}}\right)}^{-\nu }\right)}"></dd></dl> where <ul><li>d is original diameter</li> <li>Δd is rod diameter change</li> <li>ν is Poisson's ratio</li> <li>L is original length, before stretch</li> <li>ΔL is the change of length.</li></ul> The value is negative because it decreases with increase of length <div class="mw-heading mw-heading2"><h2 id="Characteristic_materials">Characteristic materials</h2>[<a href="/w/index.php?title=Poisson%27s_ratio&action=edit&section=6" title="Edit section: Characteristic materials">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Isotropic">Isotropic</h3>[<a href="/w/index.php?title=Poisson%27s_ratio&action=edit&section=7" title="Edit section: Isotropic">edit</a>]</div> For a linear isotropic material subjected only to compressive (i.e. normal) forces, the deformation of a material in the direction of one axis will produce a deformation of the material along the other axis in three dimensions. Thus it is possible to generalize <a href="/wiki/Hooke%27s_Law" class="mw-redirect" title="Hooke's Law">Hooke's Law</a> (for compressive forces) into three dimensions: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\varepsilon _{xx}&={\frac {1}{E}}\left[\sigma _{xx}-\nu \left(\sigma _{yy}+\sigma _{zz}\right)\right]\\[6px]\varepsilon _{yy}&={\frac {1}{E}}\left[\sigma _{yy}-\nu \left(\sigma _{zz}+\sigma _{xx}\right)\right]\\[6px]\varepsilon _{zz}&={\frac {1}{E}}\left[\sigma _{zz}-\nu \left(\sigma _{xx}+\sigma _{yy}\right)\right]\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.9em 0.9em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>E</mi> </mfrac> </mrow> <mrow> <mo>[</mo> <mrow> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> <mo>−</mo> <mi>ν</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>y</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>z</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>E</mi> </mfrac> </mrow> <mrow> <mo>[</mo> <mrow> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>y</mi> </mrow> </msub> <mo>−</mo> <mi>ν</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>z</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>z</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>E</mi> </mfrac> </mrow> <mrow> <mo>[</mo> <mrow> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>z</mi> </mrow> </msub> <mo>−</mo> <mi>ν</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>y</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\varepsilon _{xx}&={\frac {1}{E}}\left[\sigma _{xx}-\nu \left(\sigma _{yy}+\sigma _{zz}\right)\right]\\[6px]\varepsilon _{yy}&={\frac {1}{E}}\left[\sigma _{yy}-\nu \left(\sigma _{zz}+\sigma _{xx}\right)\right]\\[6px]\varepsilon _{zz}&={\frac {1}{E}}\left[\sigma _{zz}-\nu \left(\sigma _{xx}+\sigma _{yy}\right)\right]\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f779b643eb5a3ba0f902ab59c681a1c6bb7b357" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.634ex; margin-bottom: -0.204ex; width:30.18ex; height:18.843ex;" alt="{\displaystyle {\begin{aligned}\varepsilon _{xx}&={\frac {1}{E}}\left[\sigma _{xx}-\nu \left(\sigma _{yy}+\sigma _{zz}\right)\right]\\[6px]\varepsilon _{yy}&={\frac {1}{E}}\left[\sigma _{yy}-\nu \left(\sigma _{zz}+\sigma _{xx}\right)\right]\\[6px]\varepsilon _{zz}&={\frac {1}{E}}\left[\sigma _{zz}-\nu \left(\sigma _{xx}+\sigma _{yy}\right)\right]\end{aligned}}}">[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>]</dd></dl> where: <ul><li>εxx, εyy, and εzz are <a href="/wiki/Strain_(materials_science)" class="mw-redirect" title="Strain (materials science)">strain</a> in the direction of x, y and z</li> <li>σxx, σyy, and σzz are <a href="/wiki/Stress_(physics)" class="mw-redirect" title="Stress (physics)">stress</a> in the direction of x, y and z</li> <li>E is <a href="/wiki/Young%27s_modulus" title="Young's modulus">Young's modulus</a> (the same in all directions for isotropic materials)</li> <li>ν is Poisson's ratio (the same in all directions for isotropic materials)</li></ul> these equations can be all synthesized in the following: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{ii}={\frac {1}{E}}\left[\sigma _{ii}(1+\nu )-\nu \sum _{k}\sigma _{kk}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>E</mi> </mfrac> </mrow> <mrow> <mo>[</mo> <mrow> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>ν</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>ν</mi> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>k</mi> </mrow> </msub> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{ii}={\frac {1}{E}}\left[\sigma _{ii}(1+\nu )-\nu \sum _{k}\sigma _{kk}\right]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c171da9cc07267bd75872ef5cf780786ec35054" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:32.471ex; height:7.509ex;" alt="{\displaystyle \varepsilon _{ii}={\frac {1}{E}}\left[\sigma _{ii}(1+\nu )-\nu \sum _{k}\sigma _{kk}\right]}"></dd></dl> In the most general case, also <a href="/wiki/Shear_stress" title="Shear stress">shear stresses</a> will hold as well as normal stresses, and the full generalization of Hooke's law is given by: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{ij}={\frac {1}{E}}\left[\sigma _{ij}(1+\nu )-\nu \delta _{ij}\sum _{k}\sigma _{kk}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>E</mi> </mfrac> </mrow> <mrow> <mo>[</mo> <mrow> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>ν</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>ν</mi> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>k</mi> </mrow> </msub> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{ij}={\frac {1}{E}}\left[\sigma _{ij}(1+\nu )-\nu \delta _{ij}\sum _{k}\sigma _{kk}\right]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e72a6a8028c583aa887514378ab4c22a83cd7b47" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:35.201ex; height:7.509ex;" alt="{\displaystyle \varepsilon _{ij}={\frac {1}{E}}\left[\sigma _{ij}(1+\nu )-\nu \delta _{ij}\sum _{k}\sigma _{kk}\right]}"></dd></dl> where δij is the <a href="/wiki/Kronecker_delta" title="Kronecker delta">Kronecker delta</a>. The <a href="/wiki/Einstein_notation" title="Einstein notation">Einstein notation</a> is usually adopted: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{kk}\equiv \sum _{l}\delta _{kl}\sigma _{kl}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>k</mi> </mrow> </msub> <mo>≡</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </munder> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>l</mi> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>l</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{kk}\equiv \sum _{l}\delta _{kl}\sigma _{kl}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04beb901394cc0f30b1540e182393d2bac12ef33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:15.631ex; height:5.509ex;" alt="{\displaystyle \sigma _{kk}\equiv \sum _{l}\delta _{kl}\sigma _{kl}}"></dd></dl> to write the equation simply as: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{ij}={\frac {1}{E}}\left[\sigma _{ij}(1+\nu )-\nu \delta _{ij}\sigma _{kk}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>E</mi> </mfrac> </mrow> <mrow> <mo>[</mo> <mrow> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>ν</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>ν</mi> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>k</mi> </mrow> </msub> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{ij}={\frac {1}{E}}\left[\sigma _{ij}(1+\nu )-\nu \delta _{ij}\sigma _{kk}\right]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4894d8b32e998a6ae6b95a8356d9c9ca5db2a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:29.655ex; height:5.176ex;" alt="{\displaystyle \varepsilon _{ij}={\frac {1}{E}}\left[\sigma _{ij}(1+\nu )-\nu \delta _{ij}\sigma _{kk}\right]}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Anisotropic">Anisotropic</h3>[<a href="/w/index.php?title=Poisson%27s_ratio&action=edit&section=8" title="Edit section: Anisotropic">edit</a>]</div> For anisotropic materials, the Poisson ratio depends on the direction of extension and transverse deformation <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\nu (\mathbf {n} ,\mathbf {m} )&=-E\left(\mathbf {n} \right)s_{ij\alpha \beta }n_{i}n_{j}m_{\alpha }m_{\beta }\\[4px]E^{-1}(\mathbf {n} )&=s_{ij\alpha \beta }n_{i}n_{j}n_{\alpha }n_{\beta }\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.7em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>ν</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">m</mi> </mrow> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mi>E</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> <mo>)</mo> </mrow> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> <mi>α</mi> <mi>β</mi> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> <mi>α</mi> <mi>β</mi> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\nu (\mathbf {n} ,\mathbf {m} )&=-E\left(\mathbf {n} \right)s_{ij\alpha \beta }n_{i}n_{j}m_{\alpha }m_{\beta }\\[4px]E^{-1}(\mathbf {n} )&=s_{ij\alpha \beta }n_{i}n_{j}n_{\alpha }n_{\beta }\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb3f8484175f3a97f6ff6961b005f178b7205a31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:34.889ex; height:7.509ex;" alt="{\displaystyle {\begin{aligned}\nu (\mathbf {n} ,\mathbf {m} )&=-E\left(\mathbf {n} \right)s_{ij\alpha \beta }n_{i}n_{j}m_{\alpha }m_{\beta }\\[4px]E^{-1}(\mathbf {n} )&=s_{ij\alpha \beta }n_{i}n_{j}n_{\alpha }n_{\beta }\end{aligned}}}"></dd></dl> Here ν is Poisson's ratio, E is <a href="/wiki/Young%27s_modulus" title="Young's modulus">Young's modulus</a>, n is a unit vector directed along the direction of extension, m is a unit vector directed perpendicular to the direction of extension. Poisson's ratio has a different number of special directions depending on the type of anisotropy.<a href="#cite_note-11">[11]</a><a href="#cite_note-12">[12]</a> <div class="mw-heading mw-heading3"><h3 id="Orthotropic">Orthotropic</h3>[<a href="/w/index.php?title=Poisson%27s_ratio&action=edit&section=9" title="Edit section: Orthotropic">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Orthotropic_material" title="Orthotropic material">Orthotropic material</a></div> <a href="/wiki/Orthotropic_material" title="Orthotropic material">Orthotropic materials</a> have three mutually perpendicular planes of symmetry in their material properties. An example is wood, which is most stiff (and strong) along the grain, and less so in the other directions. Then <a href="/wiki/Hooke%27s_law" title="Hooke's law">Hooke's law</a> can be expressed in <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a> form as<a href="#cite_note-Boresi-13">[13]</a><a href="#cite_note-Lekh-14">[14]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}\epsilon _{xx}\\\epsilon _{yy}\\\epsilon _{zz}\\2\epsilon _{yz}\\2\epsilon _{zx}\\2\epsilon _{xy}\end{bmatrix}}={\begin{bmatrix}{\tfrac {1}{E_{x}}}&-{\tfrac {\nu _{yx}}{E_{y}}}&-{\tfrac {\nu _{zx}}{E_{z}}}&0&0&0\\-{\tfrac {\nu _{xy}}{E_{x}}}&{\tfrac {1}{E_{y}}}&-{\tfrac {\nu _{zy}}{E_{z}}}&0&0&0\\-{\tfrac {\nu _{xz}}{E_{x}}}&-{\tfrac {\nu _{yz}}{E_{y}}}&{\tfrac {1}{E_{z}}}&0&0&0\\0&0&0&{\tfrac {1}{G_{yz}}}&0&0\\0&0&0&0&{\tfrac {1}{G_{zx}}}&0\\0&0&0&0&0&{\tfrac {1}{G_{xy}}}\\\end{bmatrix}}{\begin{bmatrix}\sigma _{xx}\\\sigma _{yy}\\\sigma _{zz}\\\sigma _{yz}\\\sigma _{zx}\\\sigma _{xy}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>y</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>z</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>z</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>x</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mfrac> </mstyle> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>x</mi> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mfrac> </mstyle> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>x</mi> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mfrac> </mstyle> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mfrac> </mstyle> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mfrac> </mstyle> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>y</mi> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mfrac> </mstyle> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>z</mi> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mfrac> </mstyle> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>z</mi> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mfrac> </mstyle> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mfrac> </mstyle> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>z</mi> </mrow> </msub> </mfrac> </mstyle> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>x</mi> </mrow> </msub> </mfrac> </mstyle> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> </mfrac> </mstyle> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>y</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>z</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>z</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>x</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}\epsilon _{xx}\\\epsilon _{yy}\\\epsilon _{zz}\\2\epsilon _{yz}\\2\epsilon _{zx}\\2\epsilon _{xy}\end{bmatrix}}={\begin{bmatrix}{\tfrac {1}{E_{x}}}&-{\tfrac {\nu _{yx}}{E_{y}}}&-{\tfrac {\nu _{zx}}{E_{z}}}&0&0&0\\-{\tfrac {\nu _{xy}}{E_{x}}}&{\tfrac {1}{E_{y}}}&-{\tfrac {\nu _{zy}}{E_{z}}}&0&0&0\\-{\tfrac {\nu _{xz}}{E_{x}}}&-{\tfrac {\nu _{yz}}{E_{y}}}&{\tfrac {1}{E_{z}}}&0&0&0\\0&0&0&{\tfrac {1}{G_{yz}}}&0&0\\0&0&0&0&{\tfrac {1}{G_{zx}}}&0\\0&0&0&0&0&{\tfrac {1}{G_{xy}}}\\\end{bmatrix}}{\begin{bmatrix}\sigma _{xx}\\\sigma _{yy}\\\sigma _{zz}\\\sigma _{yz}\\\sigma _{zx}\\\sigma _{xy}\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df46b38b0bd87d9f186577db28a94b56ef56fe1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -13.005ex; width:59.886ex; height:27.176ex;" alt="{\displaystyle {\begin{bmatrix}\epsilon _{xx}\\\epsilon _{yy}\\\epsilon _{zz}\\2\epsilon _{yz}\\2\epsilon _{zx}\\2\epsilon _{xy}\end{bmatrix}}={\begin{bmatrix}{\tfrac {1}{E_{x}}}&-{\tfrac {\nu _{yx}}{E_{y}}}&-{\tfrac {\nu _{zx}}{E_{z}}}&0&0&0\\-{\tfrac {\nu _{xy}}{E_{x}}}&{\tfrac {1}{E_{y}}}&-{\tfrac {\nu _{zy}}{E_{z}}}&0&0&0\\-{\tfrac {\nu _{xz}}{E_{x}}}&-{\tfrac {\nu _{yz}}{E_{y}}}&{\tfrac {1}{E_{z}}}&0&0&0\\0&0&0&{\tfrac {1}{G_{yz}}}&0&0\\0&0&0&0&{\tfrac {1}{G_{zx}}}&0\\0&0&0&0&0&{\tfrac {1}{G_{xy}}}\\\end{bmatrix}}{\begin{bmatrix}\sigma _{xx}\\\sigma _{yy}\\\sigma _{zz}\\\sigma _{yz}\\\sigma _{zx}\\\sigma _{xy}\end{bmatrix}}}"></dd></dl> where <ul><li>Ei is the <a href="/wiki/Young%27s_modulus" title="Young's modulus">Young's modulus</a> along axis i</li> <li>Gij is the <a href="/wiki/Shear_modulus" title="Shear modulus">shear modulus</a> in direction j on the plane whose normal is in direction i</li> <li>νij is the Poisson ratio that corresponds to a contraction in direction j when an extension is applied in direction i.</li></ul> The Poisson ratio of an orthotropic material is different in each direction (x, y and z). However, the symmetry of the stress and strain tensors implies that not all the six Poisson's ratios in the equation are independent. There are only nine independent material properties: three elastic moduli, three shear moduli, and three Poisson's ratios. The remaining three Poisson's ratios can be obtained from the relations <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\nu _{yx}}{E_{y}}}={\frac {\nu _{xy}}{E_{x}}}\,,\qquad {\frac {\nu _{zx}}{E_{z}}}={\frac {\nu _{xz}}{E_{x}}}\,,\qquad {\frac {\nu _{yz}}{E_{y}}}={\frac {\nu _{zy}}{E_{z}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>x</mi> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>x</mi> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>z</mi> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>z</mi> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>y</mi> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\nu _{yx}}{E_{y}}}={\frac {\nu _{xy}}{E_{x}}}\,,\qquad {\frac {\nu _{zx}}{E_{z}}}={\frac {\nu _{xz}}{E_{x}}}\,,\qquad {\frac {\nu _{yz}}{E_{y}}}={\frac {\nu _{zy}}{E_{z}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3643e0be73241ae506d8247a38e054e91ad207f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:44.836ex; height:5.676ex;" alt="{\displaystyle {\frac {\nu _{yx}}{E_{y}}}={\frac {\nu _{xy}}{E_{x}}}\,,\qquad {\frac {\nu _{zx}}{E_{z}}}={\frac {\nu _{xz}}{E_{x}}}\,,\qquad {\frac {\nu _{yz}}{E_{y}}}={\frac {\nu _{zy}}{E_{z}}}}"></dd></dl> From the above relations we can see that if Ex > Ey then νxy > νyx. The larger ratio (in this case νxy) is called the major Poisson ratio while the smaller one (in this case νyx) is called the minor Poisson ratio. We can find similar relations between the other Poisson ratios. <div class="mw-heading mw-heading3"><h3 id="Transversely_isotropic">Transversely isotropic</h3>[<a href="/w/index.php?title=Poisson%27s_ratio&action=edit&section=10" title="Edit section: Transversely isotropic">edit</a>]</div> <a href="/wiki/Transversely_isotropic" class="mw-redirect" title="Transversely isotropic">Transversely isotropic</a> materials have <a href="/wiki/Transversely_isotropic" class="mw-redirect" title="Transversely isotropic">a plane of isotropy</a> in which the elastic properties are isotropic. If we assume that this plane of isotropy is the yz-plane, then Hooke's law takes the form<a href="#cite_note-Tan-15">[15]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}\epsilon _{xx}\\\epsilon _{yy}\\\epsilon _{zz}\\2\epsilon _{yz}\\2\epsilon _{zx}\\2\epsilon _{xy}\end{bmatrix}}={\begin{bmatrix}{\tfrac {1}{E_{x}}}&-{\tfrac {\nu _{yx}}{E_{y}}}&-{\tfrac {\nu _{zx}}{E_{z}}}&0&0&0\\-{\tfrac {\nu _{xy}}{E_{x}}}&{\tfrac {1}{E_{y}}}&-{\tfrac {\nu _{zy}}{E_{z}}}&0&0&0\\-{\tfrac {\nu _{xz}}{E_{x}}}&-{\tfrac {\nu _{yz}}{E_{y}}}&{\tfrac {1}{E_{z}}}&0&0&0\\0&0&0&{\tfrac {1}{G_{\rm {yz}}}}&0&0\\0&0&0&0&{\tfrac {1}{G_{\rm {zx}}}}&0\\0&0&0&0&0&{\tfrac {1}{G_{\rm {xy}}}}\\\end{bmatrix}}{\begin{bmatrix}\sigma _{xx}\\\sigma _{yy}\\\sigma _{zz}\\\sigma _{yz}\\\sigma _{zx}\\\sigma _{xy}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>y</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>z</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>z</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>x</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mfrac> </mstyle> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>x</mi> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mfrac> </mstyle> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>x</mi> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mfrac> </mstyle> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mfrac> </mstyle> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mfrac> </mstyle> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>y</mi> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mfrac> </mstyle> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>z</mi> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mfrac> </mstyle> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>z</mi> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mfrac> </mstyle> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mfrac> </mstyle> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">y</mi> <mi mathvariant="normal">z</mi> </mrow> </mrow> </msub> </mfrac> </mstyle> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">z</mi> <mi mathvariant="normal">x</mi> </mrow> </mrow> </msub> </mfrac> </mstyle> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">x</mi> <mi mathvariant="normal">y</mi> </mrow> </mrow> </msub> </mfrac> </mstyle> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>y</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>z</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>z</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>x</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}\epsilon _{xx}\\\epsilon _{yy}\\\epsilon _{zz}\\2\epsilon _{yz}\\2\epsilon _{zx}\\2\epsilon _{xy}\end{bmatrix}}={\begin{bmatrix}{\tfrac {1}{E_{x}}}&-{\tfrac {\nu _{yx}}{E_{y}}}&-{\tfrac {\nu _{zx}}{E_{z}}}&0&0&0\\-{\tfrac {\nu _{xy}}{E_{x}}}&{\tfrac {1}{E_{y}}}&-{\tfrac {\nu _{zy}}{E_{z}}}&0&0&0\\-{\tfrac {\nu _{xz}}{E_{x}}}&-{\tfrac {\nu _{yz}}{E_{y}}}&{\tfrac {1}{E_{z}}}&0&0&0\\0&0&0&{\tfrac {1}{G_{\rm {yz}}}}&0&0\\0&0&0&0&{\tfrac {1}{G_{\rm {zx}}}}&0\\0&0&0&0&0&{\tfrac {1}{G_{\rm {xy}}}}\\\end{bmatrix}}{\begin{bmatrix}\sigma _{xx}\\\sigma _{yy}\\\sigma _{zz}\\\sigma _{yz}\\\sigma _{zx}\\\sigma _{xy}\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e0ac8a0efc9eb11f27fa8d85fc8f8127e1bf359" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -13.005ex; width:59.787ex; height:27.176ex;" alt="{\displaystyle {\begin{bmatrix}\epsilon _{xx}\\\epsilon _{yy}\\\epsilon _{zz}\\2\epsilon _{yz}\\2\epsilon _{zx}\\2\epsilon _{xy}\end{bmatrix}}={\begin{bmatrix}{\tfrac {1}{E_{x}}}&-{\tfrac {\nu _{yx}}{E_{y}}}&-{\tfrac {\nu _{zx}}{E_{z}}}&0&0&0\\-{\tfrac {\nu _{xy}}{E_{x}}}&{\tfrac {1}{E_{y}}}&-{\tfrac {\nu _{zy}}{E_{z}}}&0&0&0\\-{\tfrac {\nu _{xz}}{E_{x}}}&-{\tfrac {\nu _{yz}}{E_{y}}}&{\tfrac {1}{E_{z}}}&0&0&0\\0&0&0&{\tfrac {1}{G_{\rm {yz}}}}&0&0\\0&0&0&0&{\tfrac {1}{G_{\rm {zx}}}}&0\\0&0&0&0&0&{\tfrac {1}{G_{\rm {xy}}}}\\\end{bmatrix}}{\begin{bmatrix}\sigma _{xx}\\\sigma _{yy}\\\sigma _{zz}\\\sigma _{yz}\\\sigma _{zx}\\\sigma _{xy}\end{bmatrix}}}"></dd></dl> where we have used the yz-plane of isotropy to reduce the number of constants, that is, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{y}=E_{z},\qquad \nu _{xy}=\nu _{xz},\qquad \nu _{yx}=\nu _{zx}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>,</mo> <mspace width="2em" /> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>z</mi> </mrow> </msub> <mo>,</mo> <mspace width="2em" /> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>x</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>x</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{y}=E_{z},\qquad \nu _{xy}=\nu _{xz},\qquad \nu _{yx}=\nu _{zx}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e54038ceb69d1f15124f4eede93514e740c33fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:39.239ex; height:2.843ex;" alt="{\displaystyle E_{y}=E_{z},\qquad \nu _{xy}=\nu _{xz},\qquad \nu _{yx}=\nu _{zx}.}">.</dd></dl> The symmetry of the stress and strain tensors implies that <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\nu _{xy}}{E_{x}}}={\frac {\nu _{yx}}{E_{y}}},\qquad \nu _{yz}=\nu _{zy}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>x</mi> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mfrac> </mrow> <mo>,</mo> <mspace width="2em" /> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>z</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>y</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\nu _{xy}}{E_{x}}}={\frac {\nu _{yx}}{E_{y}}},\qquad \nu _{yz}=\nu _{zy}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54f98ad8a900d586f81f37508dcd160dec8db4de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.406ex; height:5.676ex;" alt="{\displaystyle {\frac {\nu _{xy}}{E_{x}}}={\frac {\nu _{yx}}{E_{y}}},\qquad \nu _{yz}=\nu _{zy}.}"></dd></dl> This leaves us with six independent constants Ex, Ey, Gxy, Gyz, νxy, νyz. However, transverse isotropy gives rise to a further constraint between Gyz and Ey, νyz which is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{yz}={\frac {E_{y}}{2\left(1+\nu _{yz}\right)}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>z</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mrow> <mn>2</mn> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>z</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{yz}={\frac {E_{y}}{2\left(1+\nu _{yz}\right)}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f292f29b6b0a51c03b303894c351a8f38d2d595e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:18.556ex; height:6.343ex;" alt="{\displaystyle G_{yz}={\frac {E_{y}}{2\left(1+\nu _{yz}\right)}}.}"></dd></dl> Therefore, there are five independent elastic material properties two of which are Poisson's ratios. For the assumed plane of symmetry, the larger of νxy and νyx is the major Poisson ratio. The other major and minor Poisson ratios are equal. <div class="mw-heading mw-heading2"><h2 id="Poisson's_ratio_values_for_different_materials">Poisson's ratio values for different materials</h2>[<a href="/w/index.php?title=Poisson%27s_ratio&action=edit&section=11" title="Edit section: Poisson's ratio values for different materials">edit</a>]</div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:SpiderGraph_PoissonRatio.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/13/SpiderGraph_PoissonRatio.gif/500px-SpiderGraph_PoissonRatio.gif" decoding="async" width="500" height="342" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/13/SpiderGraph_PoissonRatio.gif/750px-SpiderGraph_PoissonRatio.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/1/13/SpiderGraph_PoissonRatio.gif 2x" data-file-width="911" data-file-height="623" /></a><figcaption>Influences of selected <a href="/wiki/Glass" title="Glass">glass</a> component additions on Poisson's ratio of a specific base glass.<a href="#cite_note-16">[16]</a></figcaption></figure> <dl><dd><table class="wikitable sortable" style="border-collapse: collapse"> <tbody><tr bgcolor="#cccccc"> <th>Material </th> <th>Poisson's ratio </th></tr> <tr> <td><a href="/wiki/Rubber" class="mw-redirect" title="Rubber">rubber</a> </td> <td>0.4999<a href="#cite_note-polymerphysics.net-17">[17]</a> </td></tr> <tr> <td><a href="/wiki/Gold" title="Gold">gold</a> </td> <td>0.42–0.44 </td></tr> <tr> <td>saturated <a href="/wiki/Clay" title="Clay">clay</a> </td> <td>0.40–0.49 </td></tr> <tr> <td><a href="/wiki/Magnesium" title="Magnesium">magnesium</a> </td> <td>0.252–0.289 </td></tr> <tr> <td><a href="/wiki/Titanium" title="Titanium">titanium</a> </td> <td>0.265–0.34 </td></tr> <tr> <td><a href="/wiki/Copper" title="Copper">copper</a> </td> <td>0.33 </td></tr> <tr> <td><a href="/wiki/Aluminium" title="Aluminium">aluminium</a> <a href="/wiki/Alloy" title="Alloy">alloy</a> </td> <td>0.32 </td></tr> <tr> <td><a href="/wiki/Clay" title="Clay">clay</a> </td> <td>0.30–0.45 </td></tr> <tr> <td><a href="/wiki/Stainless_steel" title="Stainless steel">stainless steel</a> </td> <td>0.30–0.31 </td></tr> <tr> <td><a href="/wiki/Steel" title="Steel">steel</a> </td> <td>0.27–0.30 </td></tr> <tr> <td><a href="/wiki/Cast_iron" title="Cast iron">cast iron</a> </td> <td>0.21–0.26 </td></tr> <tr> <td><a href="/wiki/Sand" title="Sand">sand</a> </td> <td>0.20–0.455 </td></tr> <tr> <td><a href="/wiki/Concrete" title="Concrete">concrete</a> </td> <td>0.1–0.2 </td></tr> <tr> <td><a href="/wiki/Glass" title="Glass">glass</a> </td> <td>0.18–0.3 </td></tr> <tr> <td><a href="/wiki/Metallic_glasses" class="mw-redirect" title="Metallic glasses">metallic glasses</a> </td> <td>0.276–0.409<a href="#cite_note-18">[18]</a> </td></tr> <tr> <td><a href="/wiki/Foam" title="Foam">foam</a> </td> <td>0.10–0.50 </td></tr> <tr> <td><a href="/wiki/Cork_(material)" title="Cork (material)">cork</a> </td> <td>0.0 </td></tr></tbody></table></dd></dl> <dl><dd><table class="wikitable sortable" style="border-collapse: collapse"> <tbody><tr bgcolor="#cccccc"> <th>Material</th> <th>Plane of symmetry</th> <th>νxy</th> <th>νyx</th> <th>νyz</th> <th>νzy</th> <th>νzx</th> <th>νxz </th></tr> <tr> <td><a href="/wiki/Nomex" title="Nomex">Nomex</a> <a href="/wiki/Composite_honeycomb" class="mw-redirect" title="Composite honeycomb">honeycomb core</a> </td> <td>xy, ribbon in x direction </td> <td>0.49 </td> <td>0.69 </td> <td>0.01 </td> <td>2.75 </td> <td>3.88 </td> <td>0.01 </td></tr> <tr> <td><a href="/wiki/Glass_fiber" title="Glass fiber">glass fiber</a> <a href="/wiki/Epoxy_resin" class="mw-redirect" title="Epoxy resin">epoxy resin</a> </td> <td>xy </td> <td>0.29 </td> <td>0.32 </td> <td>0.06 </td> <td>0.06 </td> <td>0.32 </td></tr></tbody></table></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Negative_Poisson's_ratio_materials">Negative Poisson's ratio materials</h3>[<a href="/w/index.php?title=Poisson%27s_ratio&action=edit&section=12" title="Edit section: Negative Poisson's ratio materials">edit</a>]</div> Some materials known as <a href="/wiki/Auxetic" class="mw-redirect" title="Auxetic">auxetic</a> materials display a negative Poisson's ratio. When subjected to positive strain in a longitudinal axis, the transverse strain in the material will actually be positive (i.e. it would increase the cross sectional area). For these materials, it is usually due to uniquely oriented, hinged molecular bonds. In order for these bonds to stretch in the longitudinal direction, the hinges must ‘open’ in the transverse direction, effectively exhibiting a positive strain.<a href="#cite_note-19">[19]</a> This can also be done in a structured way and lead to new aspects in material design as for <a href="/wiki/Mechanical_metamaterials" class="mw-redirect" title="Mechanical metamaterials">mechanical metamaterials</a>. Studies have shown that certain solid wood types display negative Poisson's ratio exclusively during a compression <a href="/wiki/Creep_(deformation)" title="Creep (deformation)">creep</a> test.<a href="#cite_note-20">[20]</a><a href="#cite_note-21">[21]</a> Initially, the compression creep test shows positive Poisson's ratios, but gradually decreases until it reaches negative values. Consequently, this also shows that Poisson's ratio for wood is time-dependent during constant loading, meaning that the strain in the axial and transverse direction do not increase in the same rate. Media with engineered microstructure may exhibit negative Poisson's ratio. In a simple case auxeticity is obtained removing material and creating a periodic porous media.<a href="#cite_note-22">[22]</a> Lattices can reach lower values of Poisson's ratio,<a href="#cite_note-23">[23]</a> which can be indefinitely close to the limiting value −1 in the isotropic case.<a href="#cite_note-24">[24]</a> More than three hundred crystalline materials have negative Poisson's ratio.<a href="#cite_note-25">[25]</a><a href="#cite_note-26">[26]</a><a href="#cite_note-27">[27]</a> For example, Li, Na, K, Cu, Rb, Ag, Fe, Ni, Co, Cs, Au, Be, Ca, Zn Sr, Sb, MoS2 and others. <div class="mw-heading mw-heading2"><h2 id="Poisson_function">Poisson function</h2>[<a href="/w/index.php?title=Poisson%27s_ratio&action=edit&section=13" title="Edit section: Poisson function">edit</a>]</div> At <a href="/wiki/Finite_strain_theory" title="Finite strain theory">finite strains</a>, the relationship between the transverse and axial strains εtrans and εaxial is typically not well described by the Poisson ratio. In fact, the Poisson ratio is often considered a function of the applied strain in the large strain regime. In such instances, the Poisson ratio is replaced by the Poisson function, for which there are several competing definitions.<a href="#cite_note-28">[28]</a> Defining the transverse stretch λtrans = εtrans + 1 and axial stretch λaxial = εaxial + 1, where the transverse stretch is a function of the axial stretch, the most common are the Hencky, Biot, Green, and Almansi functions: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\nu ^{\text{Hencky}}&=-{\frac {\ln \lambda _{\text{trans}}}{\ln \lambda _{\text{axial}}}}\\[6pt]\nu ^{\text{Biot}}&={\frac {1-\lambda _{\text{trans}}}{\lambda _{\text{axial}}-1}}\\[6pt]\nu ^{\text{Green}}&={\frac {1-\lambda _{\text{trans}}^{2}}{\lambda _{\text{axial}}^{2}-1}}\\[6pt]\nu ^{\text{Almansi}}&={\frac {\lambda _{\text{trans}}^{-2}-1}{1-\lambda _{\text{axial}}^{-2}}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.9em 0.9em 0.9em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msup> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>Hencky</mtext> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ln</mi> <mo>⁡</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>trans</mtext> </mrow> </msub> </mrow> <mrow> <mi>ln</mi> <mo>⁡</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>axial</mtext> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <msup> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>Biot</mtext> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>trans</mtext> </mrow> </msub> </mrow> <mrow> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>axial</mtext> </mrow> </msub> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <msup> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>Green</mtext> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <msubsup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>trans</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mrow> <msubsup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>axial</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <msup> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>Almansi</mtext> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msubsup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>trans</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <msubsup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>axial</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </msubsup> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\nu ^{\text{Hencky}}&=-{\frac {\ln \lambda _{\text{trans}}}{\ln \lambda _{\text{axial}}}}\\[6pt]\nu ^{\text{Biot}}&={\frac {1-\lambda _{\text{trans}}}{\lambda _{\text{axial}}-1}}\\[6pt]\nu ^{\text{Green}}&={\frac {1-\lambda _{\text{trans}}^{2}}{\lambda _{\text{axial}}^{2}-1}}\\[6pt]\nu ^{\text{Almansi}}&={\frac {\lambda _{\text{trans}}^{-2}-1}{1-\lambda _{\text{axial}}^{-2}}}\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43cdecb6568c8d4bb23922f3332da8341e960b6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -14.671ex; width:21.466ex; height:30.509ex;" alt="{\displaystyle {\begin{aligned}\nu ^{\text{Hencky}}&=-{\frac {\ln \lambda _{\text{trans}}}{\ln \lambda _{\text{axial}}}}\\[6pt]\nu ^{\text{Biot}}&={\frac {1-\lambda _{\text{trans}}}{\lambda _{\text{axial}}-1}}\\[6pt]\nu ^{\text{Green}}&={\frac {1-\lambda _{\text{trans}}^{2}}{\lambda _{\text{axial}}^{2}-1}}\\[6pt]\nu ^{\text{Almansi}}&={\frac {\lambda _{\text{trans}}^{-2}-1}{1-\lambda _{\text{axial}}^{-2}}}\end{aligned}}}"></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Applications_of_Poisson's_effect">Applications of Poisson's effect</h2>[<a href="/w/index.php?title=Poisson%27s_ratio&action=edit&section=14" title="Edit section: Applications of Poisson's effect">edit</a>]</div> One area in which Poisson's effect has a considerable influence is in pressurized pipe flow. When the air or liquid inside a pipe is highly pressurized it exerts a uniform force on the inside of the pipe, resulting in a <a href="/wiki/Hoop_stress" class="mw-redirect" title="Hoop stress">hoop stress</a> within the pipe material. Due to Poisson's effect, this hoop stress will cause the pipe to increase in diameter and slightly decrease in length. The decrease in length, in particular, can have a noticeable effect upon the pipe joints, as the effect will accumulate for each section of pipe joined in series. A restrained joint may be pulled apart or otherwise prone to failure.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] Another area of application for Poisson's effect is in the realm of <a href="/wiki/Structural_geology" title="Structural geology">structural geology</a>. Rocks, like most materials, are subject to Poisson's effect while under stress. In a geological timescale, excessive erosion or sedimentation of Earth's crust can either create or remove large vertical stresses upon the underlying rock. This rock will expand or contract in the vertical direction as a direct result of the applied stress, and it will also deform in the horizontal direction as a result of Poisson's effect. This change in strain in the horizontal direction can affect or form joints and dormant stresses in the rock.<a href="#cite_note-29">[29]</a> Although <a href="/wiki/Cork_material" class="mw-redirect" title="Cork material">cork</a> was historically chosen to seal wine bottle for other reasons (including its inert nature, impermeability, flexibility, sealing ability, and resilience),<a href="#cite_note-30">[30]</a> cork's Poisson's ratio of zero provides another advantage. As the cork is inserted into the bottle, the upper part which is not yet inserted does not expand in diameter as it is compressed axially. The force needed to insert a cork into a bottle arises only from the friction between the cork and the bottle due to the radial compression of the cork. If the stopper were made of rubber, for example, (with a Poisson's ratio of about +0.5), there would be a relatively large additional force required to overcome the radial expansion of the upper part of the rubber stopper. Most car mechanics are aware that it is hard to pull a rubber hose (such as a coolant hose) off a metal pipe stub, as the tension of pulling causes the diameter of the hose to shrink, gripping the stub tightly. (This is the same effect as shown in a <a href="/wiki/Chinese_finger_trap" title="Chinese finger trap">Chinese finger trap</a>.) Hoses can more easily be pushed off stubs instead using a wide flat blade. <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Poisson%27s_ratio&action=edit&section=15" title="Edit section: See also">edit</a>]</div> <ul><li><a href="/wiki/Linear_elasticity" title="Linear elasticity">Linear elasticity</a></li> <li><a href="/wiki/Hooke%27s_law" title="Hooke's law">Hooke's law</a></li> <li><a href="/wiki/Impulse_excitation_technique" title="Impulse excitation technique">Impulse excitation technique</a></li> <li><a href="/wiki/Orthotropic_material" title="Orthotropic material">Orthotropic material</a></li> <li><a href="/wiki/Shear_modulus" title="Shear modulus">Shear modulus</a></li> <li><a href="/wiki/Young%27s_modulus" title="Young's modulus">Young's modulus</a></li> <li><a href="/wiki/Coefficient_of_thermal_expansion" class="mw-redirect" title="Coefficient of thermal expansion">Coefficient of thermal expansion</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=Poisson%27s_ratio&action=edit&section=16" title="Edit section: References">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">^</a> For soft materials, the bulk modulus (K) is typically large compared to the shear modulus (G) so that they can be regarded as incompressible, since it is easier to change shape than to compress. This results in the Young's modulus (E) being E = 3G and hence ν = 0.5.<style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFJastrzebski1959" class="citation book cs1">Jastrzebski, D. 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Proceedings of the Royal Society A. 473 (2207): 20170607. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2017RSPSA.47370607M">2017RSPSA.47370607M</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frspa.2017.0607">10.1098/rspa.2017.0607</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5719638">5719638</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/29225507">29225507</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+Royal+Society+A&rft.atitle=How+to+characterize+a+nonlinear+elastic+material%3F+A+review+on+nonlinear+constitutive+parameters+in+isotropic+finite+elasticity&rft.volume=473&rft.issue=2207&rft.pages=20170607&rft.date=2017-11-03&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC5719638%23id-name%3DPMC&rft_id=info%3Apmid%2F29225507&rft_id=info%3Adoi%2F10.1098%2Frspa.2017.0607&rft_id=info%3Abibcode%2F2017RSPSA.47370607M&rft.aulast=Mihai&rft.aufirst=L.+A.&rft.au=Goriely%2C+A.&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC5719638&rfr_id=info%3Asid%2Fen.wikipedia.org%3APoisson%27s+ratio" class="Z3988"> </li> <li id="cite_note-29"><a href="#cite_ref-29">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www3.geosc.psu.edu/~jte2/geosc465/lect18.rtf">"Lecture Notes in Structural Geology – Effective Stress"</a>. Retrieved 2019-07-03.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Lecture+Notes+in+Structural+Geology+%E2%80%93+Effective+Stress&rft_id=http%3A%2F%2Fwww3.geosc.psu.edu%2F~jte2%2Fgeosc465%2Flect18.rtf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APoisson%27s+ratio" class="Z3988"> </li> <li id="cite_note-30"><a href="#cite_ref-30">^</a> Silva, et al. <a rel="nofollow" class="external text" href="https://thematking.com/business_industry/we-aint-just-mats/cork-products/int-materials-review-2005.pdf">"Cork: properties, capabilities and applications"</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170809095822/http://thematking.com/business_industry/we-aint-just-mats/cork-products/int-materials-review-2005.pdf">Archived</a> 2017-08-09 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, Retrieved May 4, 2017 </li> </ol></div></div> <div class="mw-heading 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href="/wiki/Bulk_modulus" title="Bulk modulus">Bulk modulus</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}">)</li> <li><a href="/wiki/Young%27s_modulus" title="Young's modulus">Young's modulus</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}">)</li> <li><a href="/wiki/Lam%C3%A9_parameters" title="Lamé parameters">Lamé's first parameter</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }">)</li> <li><a href="/wiki/Shear_modulus" title="Shear modulus">Shear modulus</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G,\mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>,</mo> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G,\mu }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed0d290f164079e9704807191c18f9415457bea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.262ex; height:2.676ex;" alt="{\displaystyle G,\mu }">)</li> <li><a class="mw-selflink selflink">Poisson's ratio</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }">)</li> <li><a href="/wiki/P-wave_modulus" title="P-wave modulus">P-wave modulus</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}">)</li></ul> </div></td></tr></tbody></table></div> <table class="wikitable mw-collapsible" width="100%" style="font-size:smaller; background:white" align="center"> <tbody><tr> <th colspan="8">Conversion formulae </th></tr> <tr> <td colspan="8">Homogeneous isotropic linear elastic materials have their elastic properties uniquely determined by any two moduli among these; thus, given any two, any other of the elastic moduli can be calculated according to these formulas, provided both for 3D materials (first part of the table) and for 2D materials (second part). </td></tr> <tr> <td style="background:#F0F0FF;">3D formulae </td> <td style="text-align:center; background:#F0F0FF;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K=\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>=</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K=\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d12e227970b986c1ff219badec7438f7383bfdb6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.906ex; height:2.176ex;" alt="{\displaystyle K=\,}"> </td> <td style="text-align:center; background:#F0F0FF;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E=\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E=\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b75f04c79769676b88c45d4fef023a01294b670a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.616ex; height:2.176ex;" alt="{\displaystyle E=\,}"> </td> <td style="text-align:center; background:#F0F0FF;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda =\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo>=</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda =\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf13c116950ce78ddbc7a016c3cd804e23ea86d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.196ex; height:2.176ex;" alt="{\displaystyle \lambda =\,}"> </td> <td style="text-align:center; background:#F0F0FF;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G=\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>=</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G=\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa8a892590600356bf9957268977269f386821c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.667ex; height:2.176ex;" alt="{\displaystyle G=\,}"> </td> <td style="text-align:center; background:#F0F0FF;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu =\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> <mo>=</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu =\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b632d33a8bb2255e8a086b63953c599d3b6a718" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.073ex; height:1.676ex;" alt="{\displaystyle \nu =\,}"> </td> <td style="text-align:center; background:#F0F0FF;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M=\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>=</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M=\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/843dc939c547010c55bd7753f7c94e6996934ca4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.283ex; height:2.176ex;" alt="{\displaystyle M=\,}"> </td> <td style="text-align:center; background:#F0F0FF;">Notes </td></tr> <tr> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (K,\,E)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>K</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>E</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (K,\,E)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/996a015d617843aa00212a26839fbae210c834a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.072ex; height:2.843ex;" alt="{\displaystyle (K,\,E)}"> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {3K(3K-E)}{9K-E}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>3</mn> <mi>K</mi> <mo stretchy="false">(</mo> <mn>3</mn> <mi>K</mi> <mo>−</mo> <mi>E</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>9</mn> <mi>K</mi> <mo>−</mo> <mi>E</mi> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {3K(3K-E)}{9K-E}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cad8348883df915b9b03fb2820ec6b096640f4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:9.215ex; height:4.343ex;" alt="{\displaystyle {\tfrac {3K(3K-E)}{9K-E}}}"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {3KE}{9K-E}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>3</mn> <mi>K</mi> <mi>E</mi> </mrow> <mrow> <mn>9</mn> <mi>K</mi> <mo>−</mo> <mi>E</mi> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {3KE}{9K-E}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cdd679253fa436a6900d1cf949d909d3b09cf3c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:5.653ex; height:3.843ex;" alt="{\displaystyle {\tfrac {3KE}{9K-E}}}"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {3K-E}{6K}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>3</mn> <mi>K</mi> <mo>−</mo> <mi>E</mi> </mrow> <mrow> <mn>6</mn> <mi>K</mi> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {3K-E}{6K}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52e22f238cec78b55c3b21bb03bd0f6bc122006b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:5.653ex; height:3.843ex;" alt="{\displaystyle {\tfrac {3K-E}{6K}}}"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {3K(3K+E)}{9K-E}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>3</mn> <mi>K</mi> <mo stretchy="false">(</mo> <mn>3</mn> <mi>K</mi> <mo>+</mo> <mi>E</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>9</mn> <mi>K</mi> <mo>−</mo> <mi>E</mi> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {3K(3K+E)}{9K-E}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c924e463a20a0f63bd2f876283e3fb1a965db68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:9.215ex; height:4.343ex;" alt="{\displaystyle {\tfrac {3K(3K+E)}{9K-E}}}"> </td> <td> </td></tr> <tr> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (K,\,\lambda )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>K</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>λ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (K,\,\lambda )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/423e5f4d1c8ee39ba6de7bc5e48df6837fff232a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.652ex; height:2.843ex;" alt="{\displaystyle (K,\,\lambda )}"> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {9K(K-\lambda )}{3K-\lambda }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>9</mn> <mi>K</mi> <mo stretchy="false">(</mo> <mi>K</mi> <mo>−</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>3</mn> <mi>K</mi> <mo>−</mo> <mi>λ</mi> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {9K(K-\lambda )}{3K-\lambda }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3233ebb13df867ba5551fefa7e092733fea80c42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:8.096ex; height:4.343ex;" alt="{\displaystyle {\tfrac {9K(K-\lambda )}{3K-\lambda }}}"> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {3(K-\lambda )}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>3</mn> <mo stretchy="false">(</mo> <mi>K</mi> <mo>−</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {3(K-\lambda )}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4d4ffacd8fb2de88a240a663a13067a7c3e921f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:6.635ex; height:4.176ex;" alt="{\displaystyle {\tfrac {3(K-\lambda )}{2}}}"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {\lambda }{3K-\lambda }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>λ</mi> <mrow> <mn>3</mn> <mi>K</mi> <mo>−</mo> <mi>λ</mi> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {\lambda }{3K-\lambda }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1c70a68ad578ac446b83ae2af368a85554e2f4d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:5.356ex; height:3.843ex;" alt="{\displaystyle {\tfrac {\lambda }{3K-\lambda }}}"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3K-2\lambda \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mi>K</mi> <mo>−</mo> <mn>2</mn> <mi>λ</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3K-2\lambda \,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dccc234a5195e4cc32a106e7280ce2e082a9da43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.974ex; height:2.343ex;" alt="{\displaystyle 3K-2\lambda \,}"> </td> <td> </td></tr> <tr> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (K,\,G)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>K</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>G</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (K,\,G)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bae48568e7a818eb72621f7cd602d0effc7e3648" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.123ex; height:2.843ex;" alt="{\displaystyle (K,\,G)}"> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {9KG}{3K+G}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>9</mn> <mi>K</mi> <mi>G</mi> </mrow> <mrow> <mn>3</mn> <mi>K</mi> <mo>+</mo> <mi>G</mi> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {9KG}{3K+G}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5440c8c124cded6c55e978cbf920e76e4883a51b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:5.689ex; height:4.009ex;" alt="{\displaystyle {\tfrac {9KG}{3K+G}}}"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K-{\tfrac {2G}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>2</mn> <mi>G</mi> </mrow> <mn>3</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K-{\tfrac {2G}{3}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5272b70cff4a6a17225acf399ea27a1948b4f59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:7.856ex; height:3.843ex;" alt="{\displaystyle K-{\tfrac {2G}{3}}}"> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {3K-2G}{2(3K+G)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>3</mn> <mi>K</mi> <mo>−</mo> <mn>2</mn> <mi>G</mi> </mrow> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <mn>3</mn> <mi>K</mi> <mo>+</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {3K-2G}{2(3K+G)}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b11dec1ab89c26fc5ad5719fd2932e1dc68f4cb3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.791ex; height:4.343ex;" alt="{\displaystyle {\tfrac {3K-2G}{2(3K+G)}}}"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K+{\tfrac {4G}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>4</mn> <mi>G</mi> </mrow> <mn>3</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K+{\tfrac {4G}{3}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a71d02abd602def118ed9f9228f0f686a06448d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:7.856ex; height:3.843ex;" alt="{\displaystyle K+{\tfrac {4G}{3}}}"> </td> <td> </td></tr> <tr> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (K,\,\nu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>K</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>ν</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (K,\,\nu )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2f6df6fb9936ea5a0e15358e66802cafe9b747d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.528ex; height:2.843ex;" alt="{\displaystyle (K,\,\nu )}"> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3K(1-2\nu )\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mi>K</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mi>ν</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3K(1-2\nu )\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd567543f973005ad7031789e6219f93463e6d87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.822ex; height:2.843ex;" alt="{\displaystyle 3K(1-2\nu )\,}"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {3K\nu }{1+\nu }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>3</mn> <mi>K</mi> <mi>ν</mi> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mi>ν</mi> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {3K\nu }{1+\nu }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1194366850f7d5421ab50b6c2706e58498f7ea96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:3.99ex; height:3.843ex;" alt="{\displaystyle {\tfrac {3K\nu }{1+\nu }}}"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {3K(1-2\nu )}{2(1+\nu )}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>3</mn> <mi>K</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mi>ν</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>ν</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {3K(1-2\nu )}{2(1+\nu )}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31c8588d1514e15c703af716b1d98c94ef2ad4c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.192ex; height:4.843ex;" alt="{\displaystyle {\tfrac {3K(1-2\nu )}{2(1+\nu )}}}"> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {3K(1-\nu )}{1+\nu }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>3</mn> <mi>K</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>ν</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mi>ν</mi> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {3K(1-\nu )}{1+\nu }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c09e8d1337a766b61ba2b9948fa4af1503d2ede0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:7.37ex; height:4.343ex;" alt="{\displaystyle {\tfrac {3K(1-\nu )}{1+\nu }}}"> </td> <td> </td></tr> <tr> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (K,\,M)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>K</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>M</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (K,\,M)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bb74f97b79a7d923d7036a9006af1437a58615e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.738ex; height:2.843ex;" alt="{\displaystyle (K,\,M)}"> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {9K(M-K)}{3K+M}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>9</mn> <mi>K</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo>−</mo> <mi>K</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>3</mn> <mi>K</mi> <mo>+</mo> <mi>M</mi> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {9K(M-K)}{3K+M}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0a754879220d62e4a49b2d4c5a9a5ac9ef939a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:8.865ex; height:4.343ex;" alt="{\displaystyle {\tfrac {9K(M-K)}{3K+M}}}"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {3K-M}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>3</mn> <mi>K</mi> <mo>−</mo> <mi>M</mi> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {3K-M}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7da3dd9c26c427aaeb80a6f841626124c66dc665" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:6.124ex; height:3.676ex;" alt="{\displaystyle {\tfrac {3K-M}{2}}}"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {3(M-K)}{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>3</mn> <mo stretchy="false">(</mo> <mi>M</mi> <mo>−</mo> <mi>K</mi> <mo stretchy="false">)</mo> </mrow> <mn>4</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {3(M-K)}{4}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/322bd48e7d8cf39df6f267e9d4fb836db4a73702" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:7.404ex; height:4.176ex;" alt="{\displaystyle {\tfrac {3(M-K)}{4}}}"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {3K-M}{3K+M}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>3</mn> <mi>K</mi> <mo>−</mo> <mi>M</mi> </mrow> <mrow> <mn>3</mn> <mi>K</mi> <mo>+</mo> <mi>M</mi> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {3K-M}{3K+M}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73b822c78e943da6485611dfbdc74be5f78b15b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:6.124ex; height:3.843ex;" alt="{\displaystyle {\tfrac {3K-M}{3K+M}}}"> </td> <td style="text-align:center;"> </td> <td> </td></tr> <tr> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (E,\,\lambda )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>E</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>λ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (E,\,\lambda )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5c948c9a642e3318131878edf4a3d45bfb90f8d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.361ex; height:2.843ex;" alt="{\displaystyle (E,\,\lambda )}"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {E+3\lambda +R}{6}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>E</mi> <mo>+</mo> <mn>3</mn> <mi>λ</mi> <mo>+</mo> <mi>R</mi> </mrow> <mn>6</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {E+3\lambda +R}{6}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a38418330bce4527d86325b426550cd9e160c575" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:7.676ex; height:3.843ex;" alt="{\displaystyle {\tfrac {E+3\lambda +R}{6}}}"> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {E-3\lambda +R}{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>E</mi> <mo>−</mo> <mn>3</mn> <mi>λ</mi> <mo>+</mo> <mi>R</mi> </mrow> <mn>4</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {E-3\lambda +R}{4}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afb261c4ee6f07fd89951fa28248a90f7b9d3025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:7.676ex; height:3.676ex;" alt="{\displaystyle {\tfrac {E-3\lambda +R}{4}}}"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {2\lambda }{E+\lambda +R}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>2</mn> <mi>λ</mi> </mrow> <mrow> <mi>E</mi> <mo>+</mo> <mi>λ</mi> <mo>+</mo> <mi>R</mi> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {2\lambda }{E+\lambda +R}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2a996d0bd87fb28da723345b090392d12e01118" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:6.854ex; height:3.843ex;" alt="{\displaystyle {\tfrac {2\lambda }{E+\lambda +R}}}"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {E-\lambda +R}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>E</mi> <mo>−</mo> <mi>λ</mi> <mo>+</mo> <mi>R</mi> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {E-\lambda +R}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/372ceea38591e2c69d5087588cab00087c749abb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:6.854ex; height:3.676ex;" alt="{\displaystyle {\tfrac {E-\lambda +R}{2}}}"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R={\sqrt {E^{2}+9\lambda ^{2}+2E\lambda }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>9</mn> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>E</mi> <mi>λ</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R={\sqrt {E^{2}+9\lambda ^{2}+2E\lambda }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8fedc22d764b6af93b68ad27937582d75919d76b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.58ex; height:3.509ex;" alt="{\displaystyle R={\sqrt {E^{2}+9\lambda ^{2}+2E\lambda }}}"> </td></tr> <tr> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (E,\,G)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>E</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>G</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (E,\,G)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee52d776f2cb07c88fbac6567029158dff718e01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.833ex; height:2.843ex;" alt="{\displaystyle (E,\,G)}"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {EG}{3(3G-E)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>E</mi> <mi>G</mi> </mrow> <mrow> <mn>3</mn> <mo stretchy="false">(</mo> <mn>3</mn> <mi>G</mi> <mo>−</mo> <mi>E</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {EG}{3(3G-E)}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94fef5491daefd3b9667bfd32eb6e4b3f31a67aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.585ex; height:4.343ex;" alt="{\displaystyle {\tfrac {EG}{3(3G-E)}}}"> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {G(E-2G)}{3G-E}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>G</mi> <mo stretchy="false">(</mo> <mi>E</mi> <mo>−</mo> <mn>2</mn> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>3</mn> <mi>G</mi> <mo>−</mo> <mi>E</mi> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {G(E-2G)}{3G-E}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b592ec0480b2ca3a99ea29d9cbd0b631d777d4d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:8.055ex; height:4.509ex;" alt="{\displaystyle {\tfrac {G(E-2G)}{3G-E}}}"> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {E}{2G}}-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>E</mi> <mrow> <mn>2</mn> <mi>G</mi> </mrow> </mfrac> </mstyle> </mrow> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {E}{2G}}-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9fb70b8d93a1714d230f745d4bb0a3aa81031a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:6.953ex; height:3.676ex;" alt="{\displaystyle {\tfrac {E}{2G}}-1}"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {G(4G-E)}{3G-E}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>G</mi> <mo stretchy="false">(</mo> <mn>4</mn> <mi>G</mi> <mo>−</mo> <mi>E</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>3</mn> <mi>G</mi> <mo>−</mo> <mi>E</mi> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {G(4G-E)}{3G-E}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06fef6489910b32a86944f5fe2f5f0c1473e87c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:8.055ex; height:4.509ex;" alt="{\displaystyle {\tfrac {G(4G-E)}{3G-E}}}"> </td> <td> </td></tr> <tr> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (E,\,\nu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>E</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>ν</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (E,\,\nu )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d858acfba62fc12da4ae6fa0759b9df87c51d96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.238ex; height:2.843ex;" alt="{\displaystyle (E,\,\nu )}"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {E}{3(1-2\nu )}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>E</mi> <mrow> <mn>3</mn> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mi>ν</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {E}{3(1-2\nu )}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49073e20128a9ecf62b06e95a6e611808b4a1015" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.731ex; height:4.176ex;" alt="{\displaystyle {\tfrac {E}{3(1-2\nu )}}}"> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {E\nu }{(1+\nu )(1-2\nu )}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>E</mi> <mi>ν</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>ν</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mi>ν</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {E\nu }{(1+\nu )(1-2\nu )}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7aa443ecbbd2c92a028d26ee25863bbc1592b4b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.16ex; height:4.176ex;" alt="{\displaystyle {\tfrac {E\nu }{(1+\nu )(1-2\nu )}}}"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {E}{2(1+\nu )}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>E</mi> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>ν</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {E}{2(1+\nu )}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea91afdc4bb2364d6deb22ee5653d33451c8d7b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:5.909ex; height:4.176ex;" alt="{\displaystyle {\tfrac {E}{2(1+\nu )}}}"> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {E(1-\nu )}{(1+\nu )(1-2\nu )}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>E</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>ν</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>ν</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mi>ν</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {E(1-\nu )}{(1+\nu )(1-2\nu )}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7abf15539e84aef12ea7f6aed6ac61b939e0897a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.16ex; height:4.843ex;" alt="{\displaystyle {\tfrac {E(1-\nu )}{(1+\nu )(1-2\nu )}}}"> </td> <td> </td></tr> <tr> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (E,\,M)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>E</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>M</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (E,\,M)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/029a8efce5680f60845e7b4029cd8d714726474c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.448ex; height:2.843ex;" alt="{\displaystyle (E,\,M)}"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {3M-E+S}{6}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>3</mn> <mi>M</mi> <mo>−</mo> <mi>E</mi> <mo>+</mo> <mi>S</mi> </mrow> <mn>6</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {3M-E+S}{6}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6df880ae07d8878db46e66ec91cde699a1172bbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:8.258ex; height:3.843ex;" alt="{\displaystyle {\tfrac {3M-E+S}{6}}}"> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {M-E+S}{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>M</mi> <mo>−</mo> <mi>E</mi> <mo>+</mo> <mi>S</mi> </mrow> <mn>4</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {M-E+S}{4}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd8aa8a8f133559b802bf384ff144d2bfec3280c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:7.436ex; height:3.676ex;" alt="{\displaystyle {\tfrac {M-E+S}{4}}}"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {3M+E-S}{8}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>3</mn> <mi>M</mi> <mo>+</mo> <mi>E</mi> <mo>−</mo> <mi>S</mi> </mrow> <mn>8</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {3M+E-S}{8}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53cddf1576995522248b1a396e8a790309691870" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:8.258ex; height:3.843ex;" alt="{\displaystyle {\tfrac {3M+E-S}{8}}}"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {E-M+S}{4M}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>E</mi> <mo>−</mo> <mi>M</mi> <mo>+</mo> <mi>S</mi> </mrow> <mrow> <mn>4</mn> <mi>M</mi> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {E-M+S}{4M}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ba7b57d39482e416644c8d99a1ac18029b84711" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:7.436ex; height:3.676ex;" alt="{\displaystyle {\tfrac {E-M+S}{4M}}}"> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=\pm {\sqrt {E^{2}+9M^{2}-10EM}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>9</mn> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>10</mn> <mi>E</mi> <mi>M</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=\pm {\sqrt {E^{2}+9M^{2}-10EM}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91fb683e4525bf2f8fef0c060113d446025ec2a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.517ex; height:3.509ex;" alt="{\displaystyle S=\pm {\sqrt {E^{2}+9M^{2}-10EM}}}"> There are two valid solutions. The plus sign leads to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu \geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu \geq 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dec80f16f256f3990d91eb9966a90bbe0baee4b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.493ex; height:2.343ex;" alt="{\displaystyle \nu \geq 0}">. The minus sign leads to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu \leq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> <mo>≤</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu \leq 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/030f3e40f7ced9ec8cc73d667aa15f61f0a377e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.493ex; height:2.343ex;" alt="{\displaystyle \nu \leq 0}">. </td></tr> <tr> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\lambda ,\,G)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>λ</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>G</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\lambda ,\,G)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7bffd6d22cc67fd1a35a585b13d7b32a74ce6b1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.412ex; height:2.843ex;" alt="{\displaystyle (\lambda ,\,G)}"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda +{\tfrac {2G}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>2</mn> <mi>G</mi> </mrow> <mn>3</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda +{\tfrac {2G}{3}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfcd78f0e042c88176c6ba5c0f8dcebf89fe223a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:7.145ex; height:3.843ex;" alt="{\displaystyle \lambda +{\tfrac {2G}{3}}}"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {G(3\lambda +2G)}{\lambda +G}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>G</mi> <mo stretchy="false">(</mo> <mn>3</mn> <mi>λ</mi> <mo>+</mo> <mn>2</mn> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>λ</mi> <mo>+</mo> <mi>G</mi> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {G(3\lambda +2G)}{\lambda +G}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0994ccfa74411f35a8f0762379eddc3660bfec4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:8.58ex; height:4.509ex;" alt="{\displaystyle {\tfrac {G(3\lambda +2G)}{\lambda +G}}}"> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {\lambda }{2(\lambda +G)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>λ</mi> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <mi>λ</mi> <mo>+</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {\lambda }{2(\lambda +G)}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/569977b3b51798797b8ed7d287e828ef07cdb54a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.466ex; height:4.343ex;" alt="{\displaystyle {\tfrac {\lambda }{2(\lambda +G)}}}"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda +2G\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo>+</mo> <mn>2</mn> <mi>G</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda +2G\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f04bd3c70df3389cd672961e8ba5636aa8d5fb48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.572ex; height:2.343ex;" alt="{\displaystyle \lambda +2G\,}"> </td> <td> </td></tr> <tr> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\lambda ,\,\nu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>λ</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>ν</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\lambda ,\,\nu )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbafda646cd2c8251ec73dbe5efd428aee76544e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.818ex; height:2.843ex;" alt="{\displaystyle (\lambda ,\,\nu )}"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {\lambda (1+\nu )}{3\nu }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>λ</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>ν</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>3</mn> <mi>ν</mi> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {\lambda (1+\nu )}{3\nu }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8668d7ae1d4ba0bea79bc24b22969de2e0020b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:6.046ex; height:4.343ex;" alt="{\displaystyle {\tfrac {\lambda (1+\nu )}{3\nu }}}"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {\lambda (1+\nu )(1-2\nu )}{\nu }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>λ</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>ν</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mi>ν</mi> <mo stretchy="false">)</mo> </mrow> <mi>ν</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {\lambda (1+\nu )(1-2\nu )}{\nu }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16ba478690ce90bc61e86d5b21ef89e8d97c0223" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.119ex; height:4.009ex;" alt="{\displaystyle {\tfrac {\lambda (1+\nu )(1-2\nu )}{\nu }}}"> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {\lambda (1-2\nu )}{2\nu }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>λ</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mi>ν</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>2</mn> <mi>ν</mi> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {\lambda (1-2\nu )}{2\nu }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0ae2f9d9801984c3a421791f68ed6d206906ce4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:6.868ex; height:4.176ex;" alt="{\displaystyle {\tfrac {\lambda (1-2\nu )}{2\nu }}}"> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {\lambda (1-\nu )}{\nu }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>λ</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>ν</mi> <mo stretchy="false">)</mo> </mrow> <mi>ν</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {\lambda (1-\nu )}{\nu }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4b0e21d0522f123eb9043c4e4352bbfd0944365" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.046ex; height:4.009ex;" alt="{\displaystyle {\tfrac {\lambda (1-\nu )}{\nu }}}"> </td> <td style="text-align:center;">Cannot be used when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu =0\Leftrightarrow \lambda =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> <mo>=</mo> <mn>0</mn> <mo stretchy="false">⇔</mo> <mi>λ</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu =0\Leftrightarrow \lambda =0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7e6655bc4e2949eee61d990716a2b8af684651d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.723ex; height:2.176ex;" alt="{\displaystyle \nu =0\Leftrightarrow \lambda =0}"> </td></tr> <tr> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\lambda ,\,M)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>λ</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>M</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\lambda ,\,M)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3242e9c8067dcfac72d5c96bcec80ccda592d34e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.028ex; height:2.843ex;" alt="{\displaystyle (\lambda ,\,M)}"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {M+2\lambda }{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>M</mi> <mo>+</mo> <mn>2</mn> <mi>λ</mi> </mrow> <mn>3</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {M+2\lambda }{3}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1df1550f26280e13b63e387c950d8e73a06d1820" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:5.622ex; height:3.843ex;" alt="{\displaystyle {\tfrac {M+2\lambda }{3}}}"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {(M-\lambda )(M+2\lambda )}{M+\lambda }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>M</mi> <mo>−</mo> <mi>λ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>M</mi> <mo>+</mo> <mn>2</mn> <mi>λ</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>M</mi> <mo>+</mo> <mi>λ</mi> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {(M-\lambda )(M+2\lambda )}{M+\lambda }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94c594e157229dfa81918f2d6de968ffe4cea963" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:12.144ex; height:4.343ex;" alt="{\displaystyle {\tfrac {(M-\lambda )(M+2\lambda )}{M+\lambda }}}"> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {M-\lambda }{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>M</mi> <mo>−</mo> <mi>λ</mi> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {M-\lambda }{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba0c175995ebb2020f1b31ad369bf52374d87816" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:4.8ex; height:3.676ex;" alt="{\displaystyle {\tfrac {M-\lambda }{2}}}"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {\lambda }{M+\lambda }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>λ</mi> <mrow> <mi>M</mi> <mo>+</mo> <mi>λ</mi> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {\lambda }{M+\lambda }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6926e4029cdea679bccd9d770206e1d6ba9cc097" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:4.8ex; height:3.843ex;" alt="{\displaystyle {\tfrac {\lambda }{M+\lambda }}}"> </td> <td style="text-align:center;"> </td> <td> </td></tr> <tr> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (G,\,\nu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>G</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>ν</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (G,\,\nu )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03de031e73561f10f24638feaf69d90abe32aea5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.289ex; height:2.843ex;" alt="{\displaystyle (G,\,\nu )}"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {2G(1+\nu )}{3(1-2\nu )}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>2</mn> <mi>G</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>ν</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>3</mn> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mi>ν</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {2G(1+\nu )}{3(1-2\nu )}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7f8f5046df3cc6a7152f9fa56845d5dbce78033" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.201ex; height:4.843ex;" alt="{\displaystyle {\tfrac {2G(1+\nu )}{3(1-2\nu )}}}"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2G(1+\nu )\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>G</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>ν</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2G(1+\nu )\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ecc39d8499aa0848351a6510b116bd2dfea06d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.421ex; height:2.843ex;" alt="{\displaystyle 2G(1+\nu )\,}"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {2G\nu }{1-2\nu }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>2</mn> <mi>G</mi> <mi>ν</mi> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mi>ν</mi> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {2G\nu }{1-2\nu }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d011b18df127233873f4af03b50318509583006e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:4.63ex; height:3.843ex;" alt="{\displaystyle {\tfrac {2G\nu }{1-2\nu }}}"> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {2G(1-\nu )}{1-2\nu }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>2</mn> <mi>G</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>ν</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mi>ν</mi> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {2G(1-\nu )}{1-2\nu }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/036c5b70861f3145642c2d797f6cbf666ea26dbc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:7.201ex; height:4.343ex;" alt="{\displaystyle {\tfrac {2G(1-\nu )}{1-2\nu }}}"> </td> <td> </td></tr> <tr> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (G,\,M)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>G</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>M</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (G,\,M)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40060aa6964ead07257199a60095ec47599c9455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.499ex; height:2.843ex;" alt="{\displaystyle (G,\,M)}"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M-{\tfrac {4G}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>4</mn> <mi>G</mi> </mrow> <mn>3</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M-{\tfrac {4G}{3}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf22c9741623f0209e7585845196a3fa1ed6282a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:8.232ex; height:3.843ex;" alt="{\displaystyle M-{\tfrac {4G}{3}}}"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {G(3M-4G)}{M-G}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>G</mi> <mo stretchy="false">(</mo> <mn>3</mn> <mi>M</mi> <mo>−</mo> <mn>4</mn> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>M</mi> <mo>−</mo> <mi>G</mi> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {G(3M-4G)}{M-G}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b53b71a131246c16879e071ac283f0cf16bc255" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:9.348ex; height:4.509ex;" alt="{\displaystyle {\tfrac {G(3M-4G)}{M-G}}}"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M-2G\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>−</mo> <mn>2</mn> <mi>G</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M-2G\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4777994467bbd835a560ddf3d9b7a9bb88f6be15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.659ex; height:2.343ex;" alt="{\displaystyle M-2G\,}"> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {M-2G}{2M-2G}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>M</mi> <mo>−</mo> <mn>2</mn> <mi>G</mi> </mrow> <mrow> <mn>2</mn> <mi>M</mi> <mo>−</mo> <mn>2</mn> <mi>G</mi> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {M-2G}{2M-2G}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d95ab5446ac8be62e025bbc4461d19ad03ddb0ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:6.777ex; height:4.009ex;" alt="{\displaystyle {\tfrac {M-2G}{2M-2G}}}"> </td> <td style="text-align:center;"> </td> <td> </td></tr> <tr> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\nu ,\,M)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>ν</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>M</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\nu ,\,M)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1fb7e1e592dc27b6b7dd908dc0f1eaae129495bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.905ex; height:2.843ex;" alt="{\displaystyle (\nu ,\,M)}"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {M(1+\nu )}{3(1-\nu )}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>M</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>ν</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>3</mn> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>ν</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {M(1+\nu )}{3(1-\nu )}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b555e35103ae080362cc5ea5e71c0a5fb01dfb3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.814ex; height:4.843ex;" alt="{\displaystyle {\tfrac {M(1+\nu )}{3(1-\nu )}}}"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {M(1+\nu )(1-2\nu )}{1-\nu }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>M</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>ν</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mi>ν</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mi>ν</mi> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {M(1+\nu )(1-2\nu )}{1-\nu }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c18235e12644cbb022f23fb36c11fca9164c19ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:11.887ex; height:4.343ex;" alt="{\displaystyle {\tfrac {M(1+\nu )(1-2\nu )}{1-\nu }}}"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {M\nu }{1-\nu }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>M</mi> <mi>ν</mi> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mi>ν</mi> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {M\nu }{1-\nu }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a015af678fe0800a1b50744d82084099f15f462a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:3.808ex; height:3.676ex;" alt="{\displaystyle {\tfrac {M\nu }{1-\nu }}}"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {M(1-2\nu )}{2(1-\nu )}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>M</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mi>ν</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>ν</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {M(1-2\nu )}{2(1-\nu )}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a076de87f5e08e5bb0733e7008687fec9f88396d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.636ex; height:4.843ex;" alt="{\displaystyle {\tfrac {M(1-2\nu )}{2(1-\nu )}}}"> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"> </td> <td> </td></tr> <tr> <td style="background:#F0F0FF;">2D formulae </td> <td style="text-align:center; background:#F0F0FF;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{\mathrm {2D} }=\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>=</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K_{\mathrm {2D} }=\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a6c0547931c4c1e3cc4ac771a535617a442d98d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.123ex; height:2.509ex;" alt="{\displaystyle K_{\mathrm {2D} }=\,}"> </td> <td style="text-align:center; background:#F0F0FF;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{\mathrm {2D} }=\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>=</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{\mathrm {2D} }=\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/145ff1eddd10383e2d0bf4f6eff8b5eeedb448f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.865ex; height:2.509ex;" alt="{\displaystyle E_{\mathrm {2D} }=\,}"> </td> <td style="text-align:center; background:#F0F0FF;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{\mathrm {2D} }=\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>=</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{\mathrm {2D} }=\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70b691041358c0200dc4c170da6b8b62706e44a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.505ex; height:2.509ex;" alt="{\displaystyle \lambda _{\mathrm {2D} }=\,}"> </td> <td style="text-align:center; background:#F0F0FF;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{\mathrm {2D} }=\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>=</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{\mathrm {2D} }=\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43b00c53cda00e94658abbcd45ed156af4c1d4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.977ex; height:2.509ex;" alt="{\displaystyle G_{\mathrm {2D} }=\,}"> </td> <td style="text-align:center; background:#F0F0FF;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu _{\mathrm {2D} }=\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>=</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu _{\mathrm {2D} }=\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2eb28b6f1e836172c154325538936f09b55de993" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.299ex; height:2.009ex;" alt="{\displaystyle \nu _{\mathrm {2D} }=\,}"> </td> <td style="text-align:center; background:#F0F0FF;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{\mathrm {2D} }=\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>=</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{\mathrm {2D} }=\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebd093c0d6e6d5c2b5f88ffbd7b23c3f0164cb20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.404ex; height:2.509ex;" alt="{\displaystyle M_{\mathrm {2D} }=\,}"> </td> <td style="text-align:center; background:#F0F0FF;">Notes </td></tr> <tr> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (K_{\mathrm {2D} },\,E_{\mathrm {2D} })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace" /> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (K_{\mathrm {2D} },\,E_{\mathrm {2D} })}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6610d0252ab47240ee324d52f781214686e42b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.538ex; height:2.843ex;" alt="{\displaystyle (K_{\mathrm {2D} },\,E_{\mathrm {2D} })}"> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {2K_{\mathrm {2D} }(2K_{\mathrm {2D} }-E_{\mathrm {2D} })}{4K_{\mathrm {2D} }-E_{\mathrm {2D} }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>2</mn> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mn>2</mn> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>−</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>4</mn> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>−</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {2K_{\mathrm {2D} }(2K_{\mathrm {2D} }-E_{\mathrm {2D} })}{4K_{\mathrm {2D} }-E_{\mathrm {2D} }}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/012be07cda111d8d43c481b55fba1c8cd9aa81da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:14.594ex; height:4.676ex;" alt="{\displaystyle {\tfrac {2K_{\mathrm {2D} }(2K_{\mathrm {2D} }-E_{\mathrm {2D} })}{4K_{\mathrm {2D} }-E_{\mathrm {2D} }}}}"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {K_{\mathrm {2D} }E_{\mathrm {2D} }}{4K_{\mathrm {2D} }-E_{\mathrm {2D} }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </mrow> <mrow> <mn>4</mn> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>−</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {K_{\mathrm {2D} }E_{\mathrm {2D} }}{4K_{\mathrm {2D} }-E_{\mathrm {2D} }}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d225deea4bae144ad3163a592c84cba353a18b4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:9.246ex; height:4.509ex;" alt="{\displaystyle {\tfrac {K_{\mathrm {2D} }E_{\mathrm {2D} }}{4K_{\mathrm {2D} }-E_{\mathrm {2D} }}}}"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {2K_{\mathrm {2D} }-E_{\mathrm {2D} }}{2K_{\mathrm {2D} }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>2</mn> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>−</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </mrow> <mrow> <mn>2</mn> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {2K_{\mathrm {2D} }-E_{\mathrm {2D} }}{2K_{\mathrm {2D} }}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/886783b8c6eba83c0af80170a833041a2bc91ff3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:9.246ex; height:4.509ex;" alt="{\displaystyle {\tfrac {2K_{\mathrm {2D} }-E_{\mathrm {2D} }}{2K_{\mathrm {2D} }}}}"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {4K_{\mathrm {2D} }^{2}}{4K_{\mathrm {2D} }-E_{\mathrm {2D} }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>4</mn> <msubsup> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mrow> <mn>4</mn> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>−</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {4K_{\mathrm {2D} }^{2}}{4K_{\mathrm {2D} }-E_{\mathrm {2D} }}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce32fb675b775c5bdfca3f4b6659163a4b55c7b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:9.246ex; height:5.009ex;" alt="{\displaystyle {\tfrac {4K_{\mathrm {2D} }^{2}}{4K_{\mathrm {2D} }-E_{\mathrm {2D} }}}}"> </td> <td> </td></tr> <tr> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (K_{\mathrm {2D} },\,\lambda _{\mathrm {2D} })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace" /> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (K_{\mathrm {2D} },\,\lambda _{\mathrm {2D} })}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8e41320590a7de1bdd74d77ec4fb8c5de98c735" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.178ex; height:2.843ex;" alt="{\displaystyle (K_{\mathrm {2D} },\,\lambda _{\mathrm {2D} })}"> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {4K_{\mathrm {2D} }(K_{\mathrm {2D} }-\lambda _{\mathrm {2D} })}{2K_{\mathrm {2D} }-\lambda _{\mathrm {2D} }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>4</mn> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>−</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>2</mn> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>−</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {4K_{\mathrm {2D} }(K_{\mathrm {2D} }-\lambda _{\mathrm {2D} })}{2K_{\mathrm {2D} }-\lambda _{\mathrm {2D} }}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff62a0404edd2ddaa137a027c26dcc9f5d54aa35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; 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width:8.503ex; height:4.509ex;" alt="{\displaystyle {\tfrac {4K_{\mathrm {2D} }G_{\mathrm {2D} }}{K_{\mathrm {2D} }+G_{\mathrm {2D} }}}}"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{\mathrm {2D} }-G_{\mathrm {2D} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>−</mo> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K_{\mathrm {2D} }-G_{\mathrm {2D} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0c746a534f84ea0488149bfc6547318c71d73b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; 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width:11.26ex; height:2.509ex;" alt="{\displaystyle K_{\mathrm {2D} }+G_{\mathrm {2D} }}"> </td> <td> </td></tr> <tr> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (K_{\mathrm {2D} },\,\nu _{\mathrm {2D} })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace" /> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (K_{\mathrm {2D} },\,\nu _{\mathrm {2D} })}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e283c72d13d40002703e71729b759b3e50c537f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; 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width:7.567ex; height:4.343ex;" alt="{\displaystyle {\tfrac {2K_{\mathrm {2D} }\nu _{\mathrm {2D} }}{1+\nu _{\mathrm {2D} }}}}"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {K_{\mathrm {2D} }(1-\nu _{\mathrm {2D} })}{1+\nu _{\mathrm {2D} }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {K_{\mathrm {2D} }(1-\nu _{\mathrm {2D} })}{1+\nu _{\mathrm {2D} }}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7feea6eb18fa6e18abbb7f5156a077c877e44a0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:10.125ex; height:4.509ex;" alt="{\displaystyle {\tfrac {K_{\mathrm {2D} }(1-\nu _{\mathrm {2D} })}{1+\nu _{\mathrm {2D} }}}}"> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {2K_{\mathrm {2D} }}{1+\nu _{\mathrm {2D} }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>2</mn> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {2K_{\mathrm {2D} }}{1+\nu _{\mathrm {2D} }}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/478b33d574f890091f0231524ec05f9f8b8ee8f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:5.6ex; height:4.343ex;" alt="{\displaystyle {\tfrac {2K_{\mathrm {2D} }}{1+\nu _{\mathrm {2D} }}}}"> </td> <td> </td></tr> <tr> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (E_{\mathrm {2D} },\,G_{\mathrm {2D} })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace" /> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (E_{\mathrm {2D} },\,G_{\mathrm {2D} })}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82f4b920c8f2b88eeddad760d3092c3fe6ade926" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.392ex; height:2.843ex;" alt="{\displaystyle (E_{\mathrm {2D} },\,G_{\mathrm {2D} })}"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {E_{\mathrm {2D} }G_{\mathrm {2D} }}{4G_{\mathrm {2D} }-E_{\mathrm {2D} }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </mrow> <mrow> <mn>4</mn> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>−</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {E_{\mathrm {2D} }G_{\mathrm {2D} }}{4G_{\mathrm {2D} }-E_{\mathrm {2D} }}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d08312a061b4190727b8cf4b3374b3a487ee0cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:9.143ex; height:4.509ex;" alt="{\displaystyle {\tfrac {E_{\mathrm {2D} }G_{\mathrm {2D} }}{4G_{\mathrm {2D} }-E_{\mathrm {2D} }}}}"> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {2G_{\mathrm {2D} }(E_{\mathrm {2D} }-2G_{\mathrm {2D} })}{4G_{\mathrm {2D} }-E_{\mathrm {2D} }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>2</mn> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>−</mo> <mn>2</mn> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>4</mn> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>−</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {2G_{\mathrm {2D} }(E_{\mathrm {2D} }-2G_{\mathrm {2D} })}{4G_{\mathrm {2D} }-E_{\mathrm {2D} }}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ba3f089d5a578367ccfdde36e561d651b70db82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:14.387ex; height:4.676ex;" alt="{\displaystyle {\tfrac {2G_{\mathrm {2D} }(E_{\mathrm {2D} }-2G_{\mathrm {2D} })}{4G_{\mathrm {2D} }-E_{\mathrm {2D} }}}}"> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {E_{\mathrm {2D} }}{2G_{\mathrm {2D} }}}-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mrow> <mn>2</mn> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </mrow> </mfrac> </mstyle> </mrow> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {E_{\mathrm {2D} }}{2G_{\mathrm {2D} }}}-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bc0b53a7580b8e0c920704ed43ffc27b9120799" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:8.804ex; height:4.343ex;" alt="{\displaystyle {\tfrac {E_{\mathrm {2D} }}{2G_{\mathrm {2D} }}}-1}"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {4G_{\mathrm {2D} }^{2}}{4G_{\mathrm {2D} }-E_{\mathrm {2D} }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>4</mn> <msubsup> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mrow> <mn>4</mn> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>−</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {4G_{\mathrm {2D} }^{2}}{4G_{\mathrm {2D} }-E_{\mathrm {2D} }}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2235f7699f8a7c484ebe6ce70240ed6f08da4c9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:9.143ex; height:5.009ex;" alt="{\displaystyle {\tfrac {4G_{\mathrm {2D} }^{2}}{4G_{\mathrm {2D} }-E_{\mathrm {2D} }}}}"> </td> <td> </td></tr> <tr> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (E_{\mathrm {2D} },\,\nu _{\mathrm {2D} })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace" /> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (E_{\mathrm {2D} },\,\nu _{\mathrm {2D} })}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b77009ceec3987a55112023183a7ee301731b675" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.714ex; height:2.843ex;" alt="{\displaystyle (E_{\mathrm {2D} },\,\nu _{\mathrm {2D} })}"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {E_{\mathrm {2D} }}{2(1-\nu _{\mathrm {2D} })}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {E_{\mathrm {2D} }}{2(1-\nu _{\mathrm {2D} })}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7a7b40ff685661adb0b9dfee6bd644d70b2e914" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.701ex; height:4.509ex;" alt="{\displaystyle {\tfrac {E_{\mathrm {2D} }}{2(1-\nu _{\mathrm {2D} })}}}"> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {E_{\mathrm {2D} }\nu _{\mathrm {2D} }}{(1+\nu _{\mathrm {2D} })(1-\nu _{\mathrm {2D} })}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {E_{\mathrm {2D} }\nu _{\mathrm {2D} }}{(1+\nu _{\mathrm {2D} })(1-\nu _{\mathrm {2D} })}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/099970a35cbfe96ff5d719bf331a6d26b1a70d47" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.922ex; height:4.509ex;" alt="{\displaystyle {\tfrac {E_{\mathrm {2D} }\nu _{\mathrm {2D} }}{(1+\nu _{\mathrm {2D} })(1-\nu _{\mathrm {2D} })}}}"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {E_{\mathrm {2D} }}{2(1+\nu _{\mathrm {2D} })}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {E_{\mathrm {2D} }}{2(1+\nu _{\mathrm {2D} })}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2caca86f5b8824866bbbb6db17c6ce009ea598e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.701ex; height:4.509ex;" alt="{\displaystyle {\tfrac {E_{\mathrm {2D} }}{2(1+\nu _{\mathrm {2D} })}}}"> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {E_{\mathrm {2D} }}{(1+\nu _{\mathrm {2D} })(1-\nu _{\mathrm {2D} })}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {E_{\mathrm {2D} }}{(1+\nu _{\mathrm {2D} })(1-\nu _{\mathrm {2D} })}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2e9fe243a025d14cc19c4feb002fa281be09200" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.922ex; height:4.509ex;" alt="{\displaystyle {\tfrac {E_{\mathrm {2D} }}{(1+\nu _{\mathrm {2D} })(1-\nu _{\mathrm {2D} })}}}"> </td> <td> </td></tr> <tr> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\lambda _{\mathrm {2D} },\,G_{\mathrm {2D} })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace" /> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\lambda _{\mathrm {2D} },\,G_{\mathrm {2D} })}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e19fd95c2596c2d28dba16de09f7c918a4fc9262" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.032ex; height:2.843ex;" alt="{\displaystyle (\lambda _{\mathrm {2D} },\,G_{\mathrm {2D} })}"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{\mathrm {2D} }+G_{\mathrm {2D} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>+</mo> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{\mathrm {2D} }+G_{\mathrm {2D} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2793b8b4c3ec84a326439a25264e5b988b625799" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.642ex; height:2.509ex;" alt="{\displaystyle \lambda _{\mathrm {2D} }+G_{\mathrm {2D} }}"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {4G_{\mathrm {2D} }(\lambda _{\mathrm {2D} }+G_{\mathrm {2D} })}{\lambda _{\mathrm {2D} }+2G_{\mathrm {2D} }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>4</mn> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>+</mo> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>+</mo> <mn>2</mn> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {4G_{\mathrm {2D} }(\lambda _{\mathrm {2D} }+G_{\mathrm {2D} })}{\lambda _{\mathrm {2D} }+2G_{\mathrm {2D} }}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fadfb075fa8e027e9917dcd1ba9bc67175d7c735" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; 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width:8.888ex; height:4.509ex;" alt="{\displaystyle {\tfrac {\lambda _{\mathrm {2D} }}{\lambda _{\mathrm {2D} }+2G_{\mathrm {2D} }}}}"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{\mathrm {2D} }+2G_{\mathrm {2D} }\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>+</mo> <mn>2</mn> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{\mathrm {2D} }+2G_{\mathrm {2D} }\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5308206440fd0cc39d9baa35004ee0675fec397" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.191ex; height:2.509ex;" alt="{\displaystyle \lambda _{\mathrm {2D} }+2G_{\mathrm {2D} }\,}"> </td> <td> </td></tr> <tr> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\lambda _{\mathrm {2D} },\,\nu _{\mathrm {2D} })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace" /> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\lambda _{\mathrm {2D} },\,\nu _{\mathrm {2D} })}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5761b070cd3f5e66294b5ff948d0d5f2ce9fd73b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.354ex; height:2.843ex;" alt="{\displaystyle (\lambda _{\mathrm {2D} },\,\nu _{\mathrm {2D} })}"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {\lambda _{\mathrm {2D} }(1+\nu _{\mathrm {2D} })}{2\nu _{\mathrm {2D} }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>2</mn> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {\lambda _{\mathrm {2D} }(1+\nu _{\mathrm {2D} })}{2\nu _{\mathrm {2D} }}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/372ee40f2f39a0530c89865308f0cfe92d46a88a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:9.688ex; height:4.509ex;" alt="{\displaystyle {\tfrac {\lambda _{\mathrm {2D} }(1+\nu _{\mathrm {2D} })}{2\nu _{\mathrm {2D} }}}}"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {\lambda _{\mathrm {2D} }(1+\nu _{\mathrm {2D} })(1-\nu _{\mathrm {2D} })}{\nu _{\mathrm {2D} }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {\lambda _{\mathrm {2D} }(1+\nu _{\mathrm {2D} })(1-\nu _{\mathrm {2D} })}{\nu _{\mathrm {2D} }}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fe813ef63bd6a0a2158043a1f270e7f2736bf49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:15.731ex; height:4.343ex;" alt="{\displaystyle {\tfrac {\lambda _{\mathrm {2D} }(1+\nu _{\mathrm {2D} })(1-\nu _{\mathrm {2D} })}{\nu _{\mathrm {2D} }}}}"> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {\lambda _{\mathrm {2D} }(1-\nu _{\mathrm {2D} })}{2\nu _{\mathrm {2D} }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>2</mn> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {\lambda _{\mathrm {2D} }(1-\nu _{\mathrm {2D} })}{2\nu _{\mathrm {2D} }}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e496450f903c59f63dc9ff43185abfc8771b7cf9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:9.688ex; height:4.509ex;" alt="{\displaystyle {\tfrac {\lambda _{\mathrm {2D} }(1-\nu _{\mathrm {2D} })}{2\nu _{\mathrm {2D} }}}}"> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {\lambda _{\mathrm {2D} }}{\nu _{\mathrm {2D} }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {\lambda _{\mathrm {2D} }}{\nu _{\mathrm {2D} }}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/775d9920e7cedfb6290becbb23ccae4407dc2131" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:3.645ex; height:4.176ex;" alt="{\displaystyle {\tfrac {\lambda _{\mathrm {2D} }}{\nu _{\mathrm {2D} }}}}"> </td> <td style="text-align:center;">Cannot be used when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu _{\mathrm {2D} }=0\Leftrightarrow \lambda _{\mathrm {2D} }=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo stretchy="false">⇔</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu _{\mathrm {2D} }=0\Leftrightarrow \lambda _{\mathrm {2D} }=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abda36be11717593314a048856e008148eab22aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.259ex; height:2.509ex;" alt="{\displaystyle \nu _{\mathrm {2D} }=0\Leftrightarrow \lambda _{\mathrm {2D} }=0}"> </td></tr> <tr> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (G_{\mathrm {2D} },\,\nu _{\mathrm {2D} })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace" /> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (G_{\mathrm {2D} },\,\nu _{\mathrm {2D} })}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a15ad10fa657886cd82c74e9cc0cd8faabb72b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.825ex; height:2.843ex;" alt="{\displaystyle (G_{\mathrm {2D} },\,\nu _{\mathrm {2D} })}"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {G_{\mathrm {2D} }(1+\nu _{\mathrm {2D} })}{1-\nu _{\mathrm {2D} }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {G_{\mathrm {2D} }(1+\nu _{\mathrm {2D} })}{1-\nu _{\mathrm {2D} }}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82c30e9987de68d9409550f95a806033d6bf05b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:10.022ex; height:4.509ex;" alt="{\displaystyle {\tfrac {G_{\mathrm {2D} }(1+\nu _{\mathrm {2D} })}{1-\nu _{\mathrm {2D} }}}}"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2G_{\mathrm {2D} }(1+\nu _{\mathrm {2D} })\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2G_{\mathrm {2D} }(1+\nu _{\mathrm {2D} })\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20b78ca05d547b5b77f79626a3b1735716468790" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.957ex; height:2.843ex;" alt="{\displaystyle 2G_{\mathrm {2D} }(1+\nu _{\mathrm {2D} })\,}"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {2G_{\mathrm {2D} }\nu _{\mathrm {2D} }}{1-\nu _{\mathrm {2D} }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>2</mn> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {2G_{\mathrm {2D} }\nu _{\mathrm {2D} }}{1-\nu _{\mathrm {2D} }}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7b74885d6aa702ca6d72909e74a0172966fb8d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:7.464ex; height:4.343ex;" alt="{\displaystyle {\tfrac {2G_{\mathrm {2D} }\nu _{\mathrm {2D} }}{1-\nu _{\mathrm {2D} }}}}"> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {2G_{\mathrm {2D} }}{1-\nu _{\mathrm {2D} }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>2</mn> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {2G_{\mathrm {2D} }}{1-\nu _{\mathrm {2D} }}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b6109dc44b8f000901eb7cd762c273aca1e2571" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:5.6ex; height:4.343ex;" alt="{\displaystyle {\tfrac {2G_{\mathrm {2D} }}{1-\nu _{\mathrm {2D} }}}}"> </td> <td> </td></tr> <tr> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (G_{\mathrm {2D} },\,M_{\mathrm {2D} })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace" /> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (G_{\mathrm {2D} },\,M_{\mathrm {2D} })}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b59af291aea1f6bf2193881aa8272fabfd9d7d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.931ex; height:2.843ex;" alt="{\displaystyle (G_{\mathrm {2D} },\,M_{\mathrm {2D} })}"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{\mathrm {2D} }-G_{\mathrm {2D} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>−</mo> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{\mathrm {2D} }-G_{\mathrm {2D} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b720400365f3d72ef91064bcf17b97f5e6564cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.541ex; height:2.509ex;" alt="{\displaystyle M_{\mathrm {2D} }-G_{\mathrm {2D} }}"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {4G_{\mathrm {2D} }(M_{\mathrm {2D} }-G_{\mathrm {2D} })}{M_{\mathrm {2D} }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>4</mn> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>−</mo> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {4G_{\mathrm {2D} }(M_{\mathrm {2D} }-G_{\mathrm {2D} })}{M_{\mathrm {2D} }}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e5e0a815db624389832f101dfb77a9a3d2c92b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:13.946ex; height:4.676ex;" alt="{\displaystyle {\tfrac {4G_{\mathrm {2D} }(M_{\mathrm {2D} }-G_{\mathrm {2D} })}{M_{\mathrm {2D} }}}}"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{\mathrm {2D} }-2G_{\mathrm {2D} }\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>−</mo> <mn>2</mn> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{\mathrm {2D} }-2G_{\mathrm {2D} }\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a8e1e8ddab785855909fc6450ee784327f49fff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.09ex; height:2.509ex;" alt="{\displaystyle M_{\mathrm {2D} }-2G_{\mathrm {2D} }\,}"> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {M_{\mathrm {2D} }-2G_{\mathrm {2D} }}{M_{\mathrm {2D} }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>−</mo> <mn>2</mn> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </mrow> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {M_{\mathrm {2D} }-2G_{\mathrm {2D} }}{M_{\mathrm {2D} }}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1048dd64851e8dd14301a6894c023470ff055dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; 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Poisson's ratio - Wikipedia